The Moduli Space of Principal G 2 -Bundles and Automorphisms

Let X be a compact Riemann surface of genus g ≥ 2 and M ( G 2 ) be the moduli space of polystable principal bundles over X , the structure group of which is the simple complex Lie group of exceptional type G 2 . In this work, it is proved that the only automorphisms that M ( G 2 ) 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 G 2 . In particular, some new results that provide relations between the stability, simplicity, and irreducibility of G 2 -bundles over X have been proved in the paper. The inclusion of groups G 2 ↪ Spin ( 8 , C ) where G 2 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 ( G 2 ) → 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 ( G 2 ) in relation to M ( Spin ( 8 , C ) ) .