Fuzzy Nambu-Goldstone Physics

TL;DR

This paper develops fuzzy matrix models for two-dimensional sigma models with target spaces like complex projective spaces and Grassmannians, capturing topological features and suitable for numerical analysis.

Contribution

It introduces fuzzy analogues of continuum sigma models on S^2 with complex target spaces, preserving topological sectors in finite-dimensional matrix form.

Findings

Fuzzy models retain continuum topological features.

Models are finite-dimensional and suitable for numerical simulations.

Applicable to Grassmannians and flag manifolds.

Abstract

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)xU(1)) ~ U(N) and G/H=CP^N. More generally our methods give fuzzy versions of continuum models on S^2 when the target spaces are Grassmannians and flag manifolds described by (N+1)x(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.