The Moduli Space of Principal <i>G</i>₂-Bundles and Automorphisms

Abstract

Let X be a compact Riemann surface of genus g≥2 and M(G2) be the moduli space of polystable principal bundles over X, the structure group of which is the simple complex Lie group of exceptional type G2. In this work, it is proved that the only automorphisms that M(G2) admits are those defined as the pull-back action of an automorphism of the base curve X. The strategy followed uses specific techniques that arise from the geometry of the gauge group G2. In particular, some new results that provide relations between the stability, simplicity, and irreducibility of G2-bundles over X have been proved in the paper. The inclusion of groups G2↪Spin(8,C) where G2 is viewed as the fixed point subgroup of an order of 3 automorphisms of Spin(8,C) that lifts the triality automorphism is also considered. Specifically, this inclusion induces the forgetful map of moduli spaces of principal bundles M(G2)→M(Spin(8,C)). In the paper, it is also proved that the forgetful map is an embedding. Finally, some consequences are drawn from the results above on the geometry of M(G2) in relation to M(Spin(8,C)).

Keywords

• principal bundle
• moduli space
• <i>G</i>₂
• automorphism
• fixed point
• forgetful map