  We analyze the spectrum of the massless Dirac operator on the 3-torus
$\mathbb{T}^3$. It is known that it is possible to calculate this spectrum
explicitly, that it is symmetric about zero and that each eigenvalue has even
multiplicity. However, for a general oriented closed Riemannian 3-manifold
$(M,g)$ there is no reason for the spectrum of the massless Dirac operator to
be symmetric. Using perturbation theory, we derive the asymptotic formulae for
its eigenvalues and prove that by the perturbation of the Euclidean metric on
the 3-torus, it is possible to obtain spectral asymmetry of the massless Dirac
operator in the axisymmetric case.
