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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$\left(W,D\right)$ 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$\left(W,D\right)$.

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 $S$.

MSC:
 58J50 Spectral problems; spectral geometry; scattering theory 37K25 Relations of infinite-dimensional systems with differential geometry 32G13 Analytic moduli problems
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