Neutrino mass matrices from localization in M-theory on $G_2$ orbifold

M-theory compactified on a $G_2$ manifold with resolved $E_8$ singularity is a promising candidate for a unified theory. The experimentally observed masses of quarks and charged leptons put a restriction on the moduli of the $G_2$ manifold. These moduli in turn uniquely determine the Dirac interactions of the neutrinos. In the paper, we explicitly compute the Dirac terms for neutrino mass matrix using the moduli from a localized model with resolved $E_8$ singularities on a $G_2$ manifold. This is a novel approach as the Dirac terms are not assumed but derived from the structure of quarks' and charged leptons' masses. Using known mass splittings and mixing angles of neutrinos, we show the acceptable region for Majorana terms. We also analyse the theoretical region for Majorana terms induced from the expectation values of right handed neutrinos through the Kolda-Martin mechanism. The intersection of the two regions indicates a restriction on neutrino masses. In particular, the lightest neutrino must have small but non-zero mass. Moreover, this also puts constraints on possible Majorana contributions from K\"ahler potential and superpotential, which can be traced down to a restriction on the geometry.We conclude that the masses of the two heavier light neutrinos are about $0.05 \text{ eV}$ and $0.009 \text{ eV}$ ($0.05 \text{ eV}$ and $0.05 \text{ eV}$)) for normal (inverted) hierarchy. In both hierarchies, we predict the light neutrinos are mostly Dirac type. Hence neutrino-less double-beta decay will be small. This is a testable result in a near future. Some bounds on heavy neutrinos are also derived.