Dirac Operators and Conformal Invariants of Tori in 3-Space

It is proved that the multipliers of the Floquet functions that are associated with immersions of tori into $\mathbb R^3$ (or $S^3$) form a complex curve in $\mathbb C^2$. The properties of this curve are studied. In addition, it is shown how the curve and its construction are related to the method of finite-gap integration, the Willmore functional, and harmonic mappings of the 2-torus into $S^3$.