  Tight-binding models can accurately predict the band structure and topology
of crystalline systems and they have been heavily used in solid-state physics
due to their versatility and low computational cost. It is quite
straightforward to build an accurate tight-binding model of any crystalline
system using the maximally localized Wannier functions of the crystal as a
basis. In 1D and 2D photonic crystals, it is possible to express the wave
equation as two decoupled scalar eigenproblems where finding a basis of
maximally localized Wannier functions is feasible using standard Wannierization
methods. Unfortunately, in 3D photonic crystals, the vectorial nature of the
electromagnetic solutions cannot be avoided. This precludes the construction of
a basis of maximally localized Wannier functions via usual techniques. In this
work, we show how to overcome this problem by using topological quantum
chemistry which will allow us to express the band structure of the photonic
crystal as a difference of elementary band representations. This can be
achieved by the introduction of a set of auxiliary modes, as recently proposed
by Solja\v{c}i\'c et. al., which regularize the $\Gamma$-point obstruction
arising from transversality constraint of the Maxwell equations. The
decomposition into elementary band representations allows us to isolate a set
of pseudo-orbitals that permit us to construct an accurate
transversality-enforced tight-binding model (TETB) that matches the dispersion,
symmetry content, and topology of the 3D photonic crystal under study.
Moreover, we show how to introduce the effects of a gyrotropic bias in the
framework, modeled via non-minimal coupling to a static magnetic field. Our
work provides the first systematic method to analytically model the photonic
bands of the lowest transverse modes over the entire BZ via a TETB model.
