Pseudoparticles and massless fermions in two dimensions

The role of pseudoparticles in the breakdown of chiral $\mathrm{U}\mathrm{}\left(N\right)\mathrm{}$ symmetry is studied in a two-dimensional model. Chiral U(1) is always destroyed by the axial-vector anomaly. For $N = 2$ chiral $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ is also spontaneously broken yielding massive fermions and three (decoupled) Goldstone bosons. For $N\ge 3$ the fermions remain massless. Realistic four-dimensional theories are believed to behave in a similar way but the critical $N$ above which the fermions cease to be massive is not known in four dimensions.