Generalised pp-waves with purely axial and purely tensor torsion of parallel Ricci curvature have their particular physical interpretation. The spinor field which completely determines the complexified curvatures of those spacetimes, satisfies the massless Dirac equation.

We consider the massless Dirac operator on a $3$-manifold which describes a single massless neutrino living in a 3-dimensional compact universe. The eigenvalues of the massless Dirac operator are interpreted to be the energy levels of that massless particle.

We break the spectral symmetry  of the massless Dirac operator on a $3$-torus using the perturbation of the Euclidean metric.

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