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

Álvaro Antón-Sancho ()
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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

Mathematics, 2024, vol. 12, issue 21, 1-20

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.

Keywords: outer automorphism; triality; Higgs bundle; fixed point; Hitchin integrable system (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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