\(F_4\) 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
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 moduli space of \(E_6\)-Higgs bundles and the automorphisms of this moduli space and the author proves that the fixed points of these automorphisms can be described as certain types of \(F_4\) or \(\mathrm{PSp}(8,\mathbb{C})\)-Higgs pairs. Section 4 is devoted to the study of stability conditions for the \(F_4\) and \(\mathrm{PSp}(8,\mathbb{C})\)-Higgs pairs. The author gives the reduced notions of stability, semistability, and polystability conditions for the families of Higgs pairs with structure group \(F_4\), which is done by using the theory set out in Section 2. In Section 5, the author discusses briefly the results obtained in this paper.
scientific article; zbMATH DE number 7643013
scientific article