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Spin(8, C )-Higgs Bundles and the Hitchin Integrable System

Author

Listed:
  • Álvaro Antón-Sancho

    (Department of Mathematics and Experimental Science, Fray Luis de Leon University College, Catholic University of Ávila, C/Tirso de Molina, 44, 47010 Valladolid, Spain
    Technology, Instruction and Design in Engineering and Education Research Group (TiDEE.rg), Catholic University of Ávila, C/Canteros, s/n, 05005 Ávila, Spain)

Abstract

Let M ( Spin ( 8 , C ) ) be the moduli space of Spin ( 8 , C ) -Higgs bundles over a compact Riemann surface X of genus g ≥ 2 . This admits a system called the Hitchin integrable system, induced by the Hitchin map, the fibers of which are Prym varieties. Moreover, the triality automorphism of Spin ( 8 , C ) acts on M ( Spin ( 8 , C ) ) , and those Higgs bundles that admit a reduction in the structure group to G 2 are fixed points of this action. This defines a map of moduli spaces of Higgs bundles M ( G 2 ) → M ( Spin ( 8 , C ) ) . In this work, the action of triality automorphism is extended to an action on the Hitchin integrable system associated with M ( Spin ( 8 , C ) ) . In particular, it is checked that the map M ( G 2 ) → M ( Spin ( 8 , C ) ) is restricted to a map at the level of the Prym varieties induced by the Hitchin map. Necessary and sufficient conditions are also provided for the Prym varieties associated with the moduli spaces of G 2 and Spin ( 8 , C ) -Higgs bundles to be disconnected. Finally, some consequences are drawn from the above results in relation to the geometry of the Prym varieties involved.

Suggested Citation

  • Álvaro Antón-Sancho, 2024. "Spin(8, C )-Higgs Bundles and the Hitchin Integrable System," Mathematics, MDPI, vol. 12(21), pages 1-20, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:21:p:3436-:d:1513047
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