Symmetry and Chern numbers:
Does the linearization method work well ?

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The Hamiltonian to be treated is a 3 by 3 traceless Hermitian matrix defined on the two-sphere with control parameters, which is assumed to be invariant under the octahedral group action. Degeneracy points are points of the two-sphere on which the Hamiltonian has degenerate eigenvalues for special parameter values. Those parameter values may form curves in the parameter space, which we call degeneracy curve. For a point away from the degeneracy curve in the parameter space, an eigen-line bundle over the sphere is associated with each non-degenerate eigenvalue. When a curve crosses the degeneracy curve in the parameter space, the associated eigen-line bundle ``after the crossing" is expected to be different from that ``before the crossing." A question is as to if one can tell the change in Chern numbers for the eigen-line bundles by studying the linearized Hamiltonian at a corresponding degeneracy point on the sphere. This talk is an interim report of the study on this theme.