Spin(8,C)-Higgs pairs over a compact Riemann surface

Abstract

Let XX be a compact Riemann surface of genus g≥2g\ge 2, GG be a semisimple complex Lie group and ρ:G→GL(V)\rho :G\to {\rm{GL}}\left(V) be a complex representation of GG. Given a principal GG-bundle EE over XX, a vector bundle E(V)E\left(V) whose typical fiber is a copy of VV is induced. A (G,ρ)\left(G,\rho )-Higgs pair is a pair (E,φ)\left(E,\varphi ), where EE is a principal GG-bundle over XX and φ\varphi is a holomorphic global section of E(V)⊗LE\left(V)\otimes L, LL being a fixed line bundle over XX. In this work, Higgs pairs of this type are considered for G=Spin(8,C)G={\rm{Spin}}\left(8,{\mathbb{C}}) and the three irreducible eight-dimensional complex representations which Spin(8,C){\rm{Spin}}\left(8,{\mathbb{C}}) admits. In particular, the reduced notions of stability, semistability, and polystability for these specific Higgs pairs are given, and it is proved that the corresponding moduli spaces are isomorphic, and a precise expression for the stable and not simple Higgs pairs associated with one of the three announced representations of Spin(8,C){\rm{Spin}}\left(8,{\mathbb{C}}) is described.

Keywords

• higgs pair
• higgs bundle
• spin
• riemann surface
• stability
• 14h10
• 14h60
• 57r57
• 53c10