Conformal maps from a 2-torus to the 4-sphere

TL;DR

This paper explores the structure of conformal immersions of a 2-torus into the 4-sphere, revealing a spectral curve framework that encodes the geometry and transforms of such immersions.

Contribution

It introduces a spectral curve approach to analyze conformal immersions of tori into the 4-sphere, linking Dirac operators, algebraic curves, and Darboux transforms.

Findings

Spectral curve arises as zero locus of Dirac operator determinant.

Kernel line bundle describes Darboux transforms.

Finite genus spectral curves lead to algebraic curves in projective space.

Abstract

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.