Quaternionic splitting of Higgs bundles for the complex symplectic group

It is well known that a Higgs bundle (E,φ) over a closed Riemann surface X for the real symplectic group Sp(2n,R) splits as the direct sum E = V⊕V*, and that φ is an antidiagonal 2×2 matrix with values β and γ on the antidiagonal. Here, V is a holomorphic rank n vector bundle, β ∈ H⁰(X, S²V ⊗ K), γ ∈ H⁰(X, S²V* ⊗ K), and K is the canonical bundle of X. The stability condition of (E,φ) can also be interpreted in terms of the bundle V and the sections β and γ.

In this talk, I will introduce a similar splitting of Higgs bundles for the complex symplectic group Sp(2n,C). We identify an Sp(2n,C)-Higgs bundle over X with a rank n quaternionic holomorphic bundle over X equipped with a certain section that corresponds to the Higgs field. This is joint work in progress with P. Huang, G. Kydonakis, and A. Wienhard.