Generalizations of parabolic Higgs bundles, real structures, and integrability

We introduce a notion of quasi-antisymmetric Higgs G-bundles over curves with marked points. They are endowed with additional structures that replace the parabolic structures at marked points in parabolic Higgs bundles. This means that the coadjoint orbits are attached to the marked points of the curves. The moduli spaces of parabolic Higgs bundles are the phase spaces of complex completely integrable systems. In our case, the coadjoint orbits are replaced by bundles cotangent to some special symmetric spaces in such a way that the moduli space of the modified Higgs bundles are still phase spaces of complex completely integrable systems. We show that the moduli space of parabolic Higgs bundles is the symplectic quotient of the moduli space of the quasi-antisymmetric Higgs bundle with respect to the action of the product of Cartan subgroups. In addition, by changing the symmetric spaces, we introduce quasi-compact and quasi-normal Higgs bundles. The fixed point sets of real involutions acting on their moduli spaces are the phase spaces of real completely integrable systems. Several examples are given including integrable extensions of the SL(2) Euler-Arnold top, two-body elliptic Calogero-Moser system, and the rational SL(2) Gaudin system together with its real reductions. Published under an exclusive license by AIP Publishing

Automated summary

The paper introduces the notion of quasi-antisymmetric Higgs G-bundles over curves with marked points, replacing parabolic structures at marked points in parabolic Higgs bundles. By modifying the coadjoint orbits, the moduli space of the modified Higgs bundles remains the phase spaces of complex completely integrable systems. The paper also explores the symplectic quotient of the moduli space, introdues quasi-compact and quasi-normal Higgs bundles, and provides examples of integrable systems.