Compactifying the rank 2 Hitchin system via spectral data on semistable curves

Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 . The geometry of the moduli space ℳ ( Spin ( 8 , C ) ) {\mathcal{ {\mathcal M} }}\left({\rm{Spin}}\left(8,{\mathbb{C}})) of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) -Higgs bundles over X X is of great interest both in algebraic geometry and mathematical physics. Consequently, several works have studied subvarieties of fixed points of this moduli space, especially those arising from the automorphism induced by the action of triality. In this work, fixed points of automorphisms of ℳ ( Spin ( 8 , C ) ) {\mathcal{ {\mathcal M} }}\left({\rm{Spin}}\left(8,{\mathbb{C}})) induced by outer automorphisms of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) are studied. This is done by giving explicit descriptions of the spectral data of these fixed points induced by the Hitchin fibration, under certain technical conditions that must be required. Specifically, it is proved that stable Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) -Higgs bundles that admit nontrivial automorphisms reduce their structure group to a subgroup isomorphic to SL ( 2 , C ) 4 = SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) {\rm{SL}}{\left(2,{\mathbb{C}})}^{4}={\rm{SL}}\left(2,{\mathbb{C}})\times {\rm{SL}}\left(2,{\mathbb{C}})\times {\rm{SL}}\left(2,{\mathbb{C}})\times {\rm{SL}}\left(2,{\mathbb{C}}) . Subsequently, the spectral data of these reductions are described and the manner in which the outer automorphisms of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) act on them is analyzed. The final application of these results allows a description of the mentioned fixed points through their spectral data.

In this research, we study the geometry of the moduli space of Spin(8,C)-Higgs bundles over a compact Riemann surface through the analysis of singular spectral curves and the triality automorphism of Spin(8,C). We establish a characterization of triality invariance, proving that a Spin(8,C)-Higgs bundle admits a reduction to the exceptional group G2 if and only if its spectral curve is invariant under the induced triality action. This transforms the problem of detecting G2-structures into a question about spectral data. We decompose the discriminant locus of the Hitchin fibration into two disjoint strata: a fixed stratum arising from G2-Higgs bundles with singular spectral curves and a free stratum consisting of orbits of size three under triality. We prove the existence of non-abelian spectral data compatible with triality symmetry, showing that non-abelian phenomena persist in free triality orbits. To quantify symmetry breaking, we introduce a triality defect invariant, which measures the dimension of the quotient of the Prym variety by its triality-invariant sublocus, and we prove that Higgs bundles with positive defects form a Zariski open dense subset.

We give a geometric characterisation of the topological invariants associated to SO(m,m+1)-Higgs bundles through KO-theory and the Langlands correspondence between orthogonal and symplectic Hitchin systems. By defining the split orthogonal spectral data, we obtain a natural grading of the moduli space of SO(m,m+1)-Higgs bundles.