The intrinsic Lagrangian fibration on the cotangent bundle of an intersection of two quadrics coincides with the Hitchin morphism of the moduli space of twisted Spin bundles over the associated hyperelliptic curve.
Spinor-valued Higgs fields
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abstract
We investigate the geometry of holomorphic vector bundles $E$ over a Riemann surface $C$ together with a section of the endomorphism bundle tensored with $K^{1/2}$ -- a square root of the canonical bundle $K$. These parallel to some extent the various features of usual Higgs bundles, such as spectral curve constructions, but some features are radically different. We make essential use of the mod 2 index to distinguish two families of moduli spaces, and provide examples in low genus.
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Intersection of two quadrics: modular interpretation and Hitchin morphism
The intrinsic Lagrangian fibration on the cotangent bundle of an intersection of two quadrics coincides with the Hitchin morphism of the moduli space of twisted Spin bundles over the associated hyperelliptic curve.