Deformation of Matrix Geometry via Landau Level Evolution

Abstract

We propose a scheme for the construction of deformed matrix geometries using Landau models. The Landau models are practically useful tools to extract matrix geometries. The level projection method however cannot be applied straightforwardly to the Landau models on deformed manifolds, as they do not generally exhibit degenerate energy levels. We overcome this problem by exploiting the idea of spectral flow. Taking a symmetric matrix geometry as a reference point of the spectral flow, we evolve the matrix geometry by deforming the Landau model. In this process, unitarity is automatically preserved. The explicit matrix realization of the coordinates is derived mechanically even for a non-perturbative deformation. We clarify basic properties of the deformed matrix geometries through a concrete analysis of the non-relativistic and relativistic Landau models on expanding two-sphere and elongating ellipsoid. The obtained ellipsoidal matrix geometries show behaviors quantitatively different in each Landau level, but qualitatively similar to their classical counterpart. We also numerically investigate the differences between the ellipsoidal matrix geometry and the fuzzy ellipsoid.