  A conformal immersion of a 2-torus into the 4-sphere is characterized by an
auxiliary Riemann surface, its spectral curve. This complex curve encodes the
monodromies of a certain Dirac type operator on a quaternionic line bundle
associated to the immersion. The paper provides a detailed description of the
geometry and asymptotic behavior of the spectral curve. If this curve has
finite genus the Dirichlet energy of a map from a 2-torus to the 2-sphere or
the Willmore energy of an immersion from a 2-torus into the 4-sphere is given
by the residue of a specific meromorphic differential on the curve. Also, the
kernel bundle of the Dirac type operator evaluated over points on the 2-torus
linearizes in the Jacobian of the spectral curve. Those results are presented
in a geometric and self contained manner.
