F 4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E 6-Higgs bundles over a compact Riemann surface

Abstract

Abstract Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 and ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) be the moduli space of E 6 {E}_{6} -Higgs bundles over X X . We consider the automorphisms σ + {\sigma }_{+} of ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) defined by σ + ( E , φ ) = ( E ∗ , − φ t ) {\sigma }_{+}\left(E,\varphi )=\left({E}^{\ast },-{\varphi }^{t}) , induced by the action of the outer involution of E 6 {E}_{6} in ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) , and σ − {\sigma }_{-} defined by σ − ( E , φ ) = ( E ∗ , φ t ) {\sigma }_{-}\left(E,\varphi )=\left({E}^{\ast },{\varphi }^{t}) , which results from the combination of σ + {\sigma }_{+} with the involution of ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) , which consists on a change of sign in the Higgs field. In this work, we describe the fixed points of σ + {\sigma }_{+} and σ − {\sigma }_{-} , as F 4 {F}_{4} -Higgs bundles, F 4 {F}_{4} -Higgs pairs associated with the fundamental irreducible representation of F 4 {F}_{4} , and PSp ( 8 , C ) {\rm{PSp}}\left(8,{\mathbb{C}}) -Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of Sp ( 8 , C ) {\rm{Sp}}\left(8,{\mathbb{C}}) . Finally, we describe the reduced notions of semistability and polystability for these objects.