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

In this paper, we study ${\mathrm{Spin}}^{*}\left(8\right)$-Higgs bundles over compact Riemann surfaces, extending the work of Bradlow, García-Prada, and Gothen on ${\mathrm{SO}}^{*}\left(8\right)$. The group ${\mathrm{Spin}}^{*}\left(8\right)$ is exceptional among classical real forms, as its complexification $\mathrm{Spin}(8,\mathbb{C})$ admits triality, an outer automorphism of order 3, but triality does not preserve the real form ${\mathrm{Spin}}^{*}\left(8\right)$. We establish the Toledo bound $\left|\tau \right|\le 4(g-1)$ for semistable ${\mathrm{Spin}}^{*}\left(8\right)$-Higgs bundles and characterize maximal bundles through rigidity theorems. We prove that the moduli space of maximal bundles fibers over the ${\mathrm{SO}}^{*}\left(8\right)$ 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 $\tau =0$ and maximal $\left|\tau \right|$. Via the non-abelian Hodge correspondence, our results yield connectedness theorems for character varieties of surface group representations into ${\mathrm{Spin}}^{*}\left(8\right)$. We analyze how triality determines the decomposition of the isotropy representation despite not acting on the real form.