Global quaternionic Hermitian band geometry on decompactified parameter spaces

Determine whether a gapped mixed lattice-continuum Hamiltonian can induce quaternionic Hermitian band geometry globally on at least one of the parameter spaces X_a = T^{4-a} × R^a for a = 1, 2, 3, or 4.

Background

The paper establishes that a single occupied quaternionic band on the compact parameter space T^4 cannot simultaneously achieve everywhere saturation of the quaternionic Wirtinger inequality and an everywhere nondegenerate quantum metric. By contrast, the stereographic pullback of the S^4 construction realizes the desired geometry on R^4, suggesting that decompactification may remove the topological obstruction.

The authors introduce the interpolating family X_a = T^{4-a} × R^a, with a = 0, 1, 2, 3, 4. For a ≥ 1, they note that decompactification removes the compactness argument responsible for the T^4 obstruction, but emphasize that this does not guarantee a physically admissible realization. The minimally decompactified space T^3 × R is identified as a natural starting point, particularly in view of self-dual Yang–Mills configurations under suitable asymptotic or twisted boundary conditions.