preprints.org > computer science and mathematics > geometry and topology > doi: 10.20944/preprints202409.1678.v2 (registering DOI)

Preprint Article Version 2 This version is not peer-reviewed

Version 1 : Received: 20 September 2024 / Approved: 21 September 2024 / Online: 24 September 2024 (03:49:20 CEST)
Version 2 : Received: 28 October 2024 / Approved: 28 October 2024 / Online: 29 October 2024 (01:51:04 CET)

Let $\mathcal{M}(\text{Spin}(8,\mathbb{C}))$ be the moduli space of $\text{Spin}(8,\mathbb{C})$-Higgs bundles over a compact Riemann surface $X$ of genus $g\geq 2$. It admits a system, called Hitchin integrable system, induced by the Hitchin map, whose fibers are Prym varieties. Also, the triality automorphism of $\text{Spin}(8,\mathbb{C})$ acts on $\mathcal{M}(\text{Spin}(8,\mathbb{C}))$ and those Higgs bundles that admit a reduction of structure group to $G_2$ are fixed points of this action. This defines a map of moduli spaces of Higgs bundles $\mathcal{M}(G_2)\rightarrow\mathcal{M}(\text{Spin}(8,\mathbb{C}))$. In this work, the action of the triality automorphism is extended to an action on the Hitchin integrable system associated to $\mathcal{M}(\text{Spin}(8,\mathbb{C}))$. In particular, it is checked that the map $\mathcal{M}(G_2)\rightarrow\mathcal{M}(\text{Spin}(8,\mathbb{C}))$ restricts 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 $\text{Spin}(8,\mathbb{C})$-Higgs bundles to be disconnected. Finally, some consequences are drawn from the above results in relation to the geometry of the Prym varieties involved.

outer automorphism; triality; Higgs bundle; fixed point; Hitchin integrable system

Computer Science and Mathematics, Geometry and Topology

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