Cyclic Higgs bundles and the Toledo invariant

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 $\mathfrak{g}={⨁}_{j=1-m}^{m-1}{\mathfrak{g}}_j$ by using the natural representation $G_0\to GL({\mathfrak{g}}_1\oplus {\mathfrak{g}}_{1-m})$, where $G_0\le G$ is the connected subgroup corresponding to ${\mathfrak{g}}_0$. The resulting Higgs pairs include $G^{\mathbb{R}}$-Higgs bundles for $G^{\mathbb{R}}\le G$ a real form of Hermitian type (in the case $m=2$) and fixed points of the ${\mathbb{C}}^{\ast }$-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.