Fractional quantum numbers, complex orbifolds and noncommutative geometry

This paper studies the conductance on the universal homology covering space Z of 2D orbifolds in a strong magnetic field, thereby removing the rationality constraint on the magnetic field in earlier works (Avron et al 1994 Phys. Rev. Lett. 73 3255-3257; Mathai and Wilkin 2019 Lett. Math. Phys. 109 2473-2484; Prieto 2006 Commun. Math. Phys. 265 373-396) in the literature. We consider a natural Landau Hamiltonian on Z and study its spectrum which we prove consists of a finite number of low-lying isolated points and calculate the von Neumann degree of the associated holomorphic spectral orbibundles when the magnetic field B is large, and obtain fractional quantum numbers as the conductance.