Cyclic Higgs Bundles and the Toledo Invariant

Abstract Let G be a complex semisimple Lie group and $$\mathfrak {g}$$ its Lie algebra. In this paper, we study a special class of cyclic Higgs bundles constructed from a $$\mathbb {Z}$$ -grading $$\mathfrak {g}= \bigoplus _{j=1-m}^{m-1}\mathfrak {g}_j$$ by using the natural representation $$G_0 \rightarrow {{\,\textrm{GL}\,}}(\mathfrak {g}_1 \oplus \mathfrak {g}_{1-m})$$ , where $$G_0 \leqslant G$$ is the connected subgroup corresponding to $$\mathfrak {g}_0$$ . The resulting Higgs pairs include $$G^\mathbb {R}$$ -Higgs bundles for $$G^\mathbb {R}\leqslant G$$ a real form of Hermitian type (in the case $$m=2$$ ) and fixed points of the $$\mathbb {C}^*$$ -action on G -Higgs bundles (in the case where the Higgs field vanishes along $$\mathfrak {g}_{1-m}$$ ). In both of these situations a topological invariant with interesting properties, known as the Toledo invariant, has been defined and studied in the literature. This paper generalises its definition and properties to the case of arbitrary $$(G_0,\mathfrak {g}_1 \oplus \mathfrak {g}_{1-m})$$ -Higgs pairs, which give rise to families of cyclic Higgs bundles. The results are applied to the example with $$m=3$$ that arises from the theory of quaternion-Kähler symmetric spaces.