Topology vs localization in synthetic dimensions

Motivated by recent developments in quantum simulation of synthetic dimensions, e.g., optical lattices ultracold atoms, we discuss here d-dimensional periodic, gapped systems for d ≤ 4, with a focus on the topology occupied energy states. We perform this analysis asking whether spectral subspace below gap can be spanned smooth and periodic Bloch functions, corresponding to localized Wannier functions position space. By constructing these inductively dimension, show that if they are required orthonormal, then, general, their existence is obstructed first two Chern classes underlying bundle, second class characterizing, particular, four-dimensional situation. If orthonormality constraint relaxed, how m bands Parseval frame comprising at most + 2 functions.