\(F_4\) and PSp\((8,\mathbb{C})\)-Higgs pairs understood as fixed points of the moduli space of \(E_6\)-Higgs bundles over a compact Riemann surface
type: Item
instance of: scholarly article
title: \(F_4\) and PSp\((8,\mathbb{C})\)-Higgs pairs understood as fixed points of the moduli space of \(E_6\)-Higgs bundles over a compact Riemann surface
zbMATH Open document ID: 1504.14018
published in: Open Mathematics
publication date: 2023-01-17T00:00:00Z
review text: This paper deals with a family of Higgs pairs over a compact Riemann surface \(X\) of genus \(g\geq2\), whose structure group is the exceptional simple complex Lie group \(F_4\) or the classical complex Lie group \(\mathrm{PSp}(8,\mathbb{C})\), and the centerless group whose universal cover is \(\mathrm{Sp}(8,\mathbb{C})\). The author considers the automorphisms \(\sigma_+\) of the moduli space \(\mathcal{M}(E_6)\) of \(E_6\)-Higgs bundles over \(X\), defined by \(\sigma_+(E,\varphi)=(E^*,-\varphi^t)\) induced by the action of the outer involution of \(E_6\) in \(\mathcal{M}(E_6)\), and \(\sigma_-\) defined by \(\sigma_-(E,\varphi)=(E^*,\varphi^t)\), which results from the combination of \(\sigma_+\) with the involution of \(\mathcal{M}(E_6)\), which consists on a change of sign in the Higgs field. The main result of this paper describes the subvariety of fixed points of \(\mathcal{M}(E_6)\) for these two automorphisms. The author describes the fixed points of \(\sigma_+\) and \(\sigma_-\), as \(F_4\)-Higgs bundles, \(F_4\)-Higgs pairs associated with the fundamental irreducible representation of \(F_4\), and \(\mathrm{PSp}(8,\mathbb{C})\)-Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of \(\mathrm{Sp}(8,\mathbb{C})\). He gives in a concrete description of the reduced notions of semistability and polystability for these objects. This paper is organized as follows: Section 1 is an introduction to the subject and summarizes the main results. In Section 2, the author introduces the concept of Higgs pair and the corresponding stability conditions needed to define the moduli space of Higgs pairs over a compact Riemann surface \(X\) of genus \(g\geq2\) and with a given complex semi-simple structure group \(G\) as set out in [\textit{O. Garcia-Prada}, et al., ``The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations'', Preprint, \url{arXiv:0909.4487}]. Section 3 deals with the m…
reviewed by: Ahmed Lesfari
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H60
zbMATH DE Number: 7643013
zbMATH Keywords: Higgs pairs
zbMATH Keywords: Lie group \(\mathrm{E}_6\)
zbMATH Keywords: automorphism
zbMATH Keywords: stability
zbMATH Keywords: fixed points
author: Álvaro Antón Sancho
MaRDI profile type: Publication
full work available at URL: math-2022-0543
OpenAlex ID: W4313403471
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DOI: 10.1515/MATH-2022-0543
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description: scientific article; zbMATH DE number 7643013