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The spectral curve of a quaternionic holomorphic line bundle over a 2-torus. (English) Zbl 1183.58026

Let W be a right quaternionic vector bundle over a 2-torus T which has a holomorphic structure D. Let Spec(W,D) be the associated spectrum of complex monodromies occuring for non-trivial holomorphic sections over the pull-back bundle on the universal cover. This is a 1-dimensional complex analytic set. The associated spectral curve S is the Riemann surface which normalizes Spec(W,D).

The authors provide a detailed desciption of the geometry and asymptotic behavior of S. If S has finite genus, the Dirichlet energy of a map from T 2 to S 2 or the Willmore energy of an immersion of T 2 into S 4 are presented in terms of the geometry of S. Similarly, the kernel bundle of a Dirac type bundle can be related to the Jacobian of S.

58J50Spectral problems; spectral geometry; scattering theory
37K25Relations of infinite-dimensional systems with differential geometry
32G13Analytic moduli problems
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