Moduli Spaces of Framed G–Higgs Bundles and Symplectic Geometry

Abstract

Let X be a compact connected Riemann surface, D ⊂ X a reduced effective divisor, G a connected complex reductive affine algebraic group and Hx G a Zariski closed subgroup for every x ∈ D. A framed principal G–bundle on X is a pair (EG, φ), where EG is a holomorphic principal G–bundle on X and φ assigns to each x ∈ D a point of the quotient space (EG)x /Hx . A framed G–Higgs bundle is a framed principal G–bundle (EG, φ) together with a holomorphic section θ ∈ H0(X, ad(EG) ⊗ KX ⊗ OX (D)) such that θ (x) is compatible with the framing φ at x for every x ∈ D. We construct a holomorphic symplectic structure on the moduli space MF H (G) of stable framed G–Higgs bundles on X. Moreover, we prove that the natural morphism from MF H (G) to the moduli space MH (G) of D-twisted G–Higgs bundles (EG, θ) that forgets the framing, is Poisson. These results generalize (Biswas et al. in Int Math Res Not, 2019. https://doi.org/10.1093/imrn/rnz016,arXiv:1805.07265) where (G, {Hx }x∈D) is taken to be (GL(r, C), {Ir×r}x∈D). We also investigate the Hitchin system for the moduli space MF H (G) and its relationship with that for MH (G).