  We study the space of conformal immersions of a 2-torus into the 4-sphere.
The moduli space of generalized Darboux transforms of such an immersed torus
has the structure of a Riemann surface, the spectral curve. This Riemann
surface arises as the zero locus of the determinant of a holomorphic family of
Dirac type operators parameterized over the complexified dual torus. The kernel
line bundle of this family over the spectral curve describes the generalized
Darboux transforms of the conformally immersed torus. If the spectral curve has
finite genus the kernel bundle can be extended to the compactification of the
spectral curve and we obtain a linear 2-torus worth of algebraic curves in
projective 3-space. The original conformal immersion of the 2-torus is
recovered as the orbit under this family of the point at infinity on the
spectral curve projected to the 4-sphere via the twistor fibration.
