Rigidity and Toledo Invariant for Spin*(8)-Higgs Bundles

In this paper, we study Spin*(8)-Higgs bundles over compact Riemann surfaces, extending the work of Bradlow, García-Prada, and Gothen on SO*(8). The group Spin*(8) is exceptional among classical real forms, as its complexification Spin(8,C) admits triality, an outer automorphism of order 3, but triality does not preserve the real form Spin*(8). We establish the Toledo bound |τ|≤4(g−1) for semistable Spin*(8)-Higgs bundles and characterize maximal bundles through rigidity theorems. We prove that the moduli space of maximal bundles fibers over the SO*(8) moduli space with discrete fibers parametrized by spin structures, and has a dimension of 15(g−1), one less than expected. Using Morse theory, we establish connectedness of moduli spaces for τ=0 and maximal |τ|. Via the non-abelian Hodge correspondence, our results yield connectedness theorems for character varieties of surface group representations into Spin*(8). We analyze how triality determines the decomposition of the isotropy representation despite not acting on the real form.