The spectral curve of a quaternionic holomorphic line bundle over a 2-torus

TL;DR

This paper explores the spectral curve associated with quaternionic holomorphic line bundles over a 2-torus, linking spectral geometry to energy functionals of conformal immersions into spheres.

Contribution

It provides a detailed geometric analysis of the spectral curve, including its asymptotics and relation to energy functionals, for quaternionic holomorphic line bundles over 2-tori.

Findings

Spectral curve encodes monodromies of a Dirac type operator.

Finite genus spectral curves relate to energy functionals via residues.

Kernel bundle linearizes in the Jacobian of the spectral curve.

Abstract

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.