What are the building blocks of photonic band structures?

Topological quantum chemistry (TQC) classifies band topology of electronic systems based on a real-space picture with localized orbitals. However, a naive application of TQC for topological characterization of the lowest photonic bands breaks down due to the presence of a polarization singularity at zero frequency and momentum. A recent approach to regularize the symmetry properties at the singularity involves adding trivial auxiliary longitudinal modes to the band structure. Inspired by this approach, we present a general framework which uses real-space invariants to understand how photonic band structures can be built from localized orbitals for any space group. We demonstrate how this method lays bare the similarities and dissimilarities between electronic and photonic bands by considering several examples from different space groups. These claims are then verified by constructing tight-binding models and simulating photonic crystal structures for each example. Finally, we explore how the polarization singularity and its regularization constrain the Wilson loop spectrum and Wannierization of photonic bands.