Zero modes of non-abelian Dirac operator in topologically non-trivial band insulator

We show that the local gauge-invariance of the quantum geometric tensor (QGT) defined in the Block-momentum space of a generic $N$-level (sublattice degrees of freedom) band insulator implies the existence of zero modes of non-abelian Dirac operator in such momentum space. Solutions of these zero modes equations in the two-dimensional Brillouin zone torus, in terms of Jacobi Theta function determine the probability amplitudes associated with the $N$-component ground state wave-function under adiabatic approximation in this Hilbert space. These solutions subjected to normalization, defines a complex projective ($CP$) space of $N-1$ dimension ($CP^{N-1}$ space) when one or more degeneracy points exist in the dispersion spectrum of such band-isulator. We show how the non-abelian generalization of the vortexability criterion of Chern bands automatically follows from these zero-mode equations, and also demonstrate their connection with momentum space-version of Lowest landau level algebra. Subsequently we write an Euclidean action from which these zero mode equations follow. We point out that the non-interacting part of different paradigms used to understand fractional Chern insulator(FCI) like phases in a host of two-dimensional material can be understood within this approach. We analyse two effective hamiltonian : lattice Dirac (QZW) model and two-band model for rhombohedral $N$-layer graphene in our propsoed framework and obtain important conclusions.