Spin(8,C)-Higgs Bundles and the Hitchin Integrable System
Year: 2024
Type: article
Abstract: Let $\mathcal{M}(\Spin(8,\mathbb{C}))$ be the moduli space of $\Spin(8,\mathbb{C})$-Higgs bundles over a compact Riemann surface $X$ of genus $g\geq 2$. The triality automorphism of $\Spin(8,\mathbb{C})$ acts on $\mathcal{M}(\Spin(8,\mathbb{C}))$ 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 $\mathcal{M}(G_2)\rightarrow\mathcal{M}(\Spin(8,\mathbb{C}))$. In this work, the action of the triality automorphism is extended to an action on the Hitchin integrable system associated to $\mathcal{M}(\Spin(8,\mathbb{C}))$. In particular, it is checked that the map $\mathcal{M}(G_2)\rightarrow\mathcal{M}(\Spin(8,\mathbb{C}))$ restricts to a map at the level of Prym varieties. Necessary and sufficient conditions are also provided for the Prym varieties associated with the moduli spaces of $G_2$ and $\Spin(8,\mathbb{C})$-Higgs bundles to be disconnected. Finally, some consequences are drawn from the above results in relation to the geometry of the Prym varieties involved.
Source: Mathematics
DOI: 10.3390/math12213436
Author: Álvaro Antón-Sancho
Institution: Catholic University of Ávila
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,955
OpenAlex ID: W4404044330