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20 pages, 492 KiB

Open AccessFeature PaperArticle

Spin(8,

C

)-Higgs Bundles and the Hitchin Integrable System

by Álvaro Antón-Sancho

Mathematics 2024, 12(21), 3436; https://doi.org/10.3390/math12213436 - 3 Nov 2024

Cited by 7 | Viewed by 955

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 \geq 2

. This admits a system called the Hitchin integrable [...] Read more.

Let

M (Spin (8, C))

be the moduli space of

Spin (8, C)

-Higgs bundles over a compact Riemann surface X of genus

g \geq 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}) \to 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}) \to 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. Full article

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