Fuzzy Nambu-Goldstone Physics

In spacetime dimensions larger than 2, whenever a global symmetry G is spontaneously broken to a subgroup H, and G and H are Lie groups, there are Nambu -Goldstone modes described by fields with values in G/H. In two-dimensional spacetimes as well, models where fields take values in G/H are of considerable interest even though in that case there is no spontaneous breaking of continuous symmetries. We consider such models when the world sheet is a two-sphere and describe their fuzzy analogues for G = SU(N+1), H = S(U(N −1)⊗U(1)) ≃ U(N) and G/H = CPN . More generally our methods give fuzzy versions of continuum models on S2 when the target spaces are Grassmannians and flag manifolds described by (N +1)× (N +1) projectors of rank ≤ (N +1)/2. These fuzzy models are finite-dimensional matrix models which nevertheless retain all the essential continuum topological features like solitonic sectors. They seem well-suited for numerical work.