F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface

Abstract

Let XX be a compact Riemann surface of genus g≥2g\ge 2 and ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}) be the moduli space of E6{E}_{6}-Higgs bundles over XX. We consider the automorphisms σ+{\sigma }_{+} of ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}) defined by σ+(E,φ)=(E∗,−φt){\sigma }_{+}\left(E,\varphi )=\left({E}^{\ast },-{\varphi }^{t}), induced by the action of the outer involution of E6{E}_{6} in ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}), and σ−{\sigma }_{-} defined by σ−(E,φ)=(E∗,φt){\sigma }_{-}\left(E,\varphi )=\left({E}^{\ast },{\varphi }^{t}), which results from the combination of σ+{\sigma }_{+} with the involution of ℳ(E6){\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 F4{F}_{4}-Higgs bundles, F4{F}_{4}-Higgs pairs associated with the fundamental irreducible representation of F4{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.

Keywords

• higgs pairs
• lie group e6
• automorphism
• stability
• fixed points
• 14d20
• 14h10
• 14h60