Moduli Spaces of Framed G–Higgs Bundles and Symplectic Geometry

Abstract

Let X be a compact connected Riemann surface, $D\, \subset \, X$ a reduced effective divisor, G a connected complex reductive affine algebraic group and $H_x\, \subsetneq \, G$ a Zariski closed subgroup for every $x\, \in \, D$. A framed principal G–bundle on X is a pair $(E_G,\, \phi )$, where $E_G$ is a holomorphic principal G–bundle on X and $\phi$ assigns to each $x\, \in \, D$ a point of the quotient space $(E_G)_x/H_x$. A framed G–Higgs bundle is a framed principal G–bundle $(E_G,\, \phi )$ together with a holomorphic section $\theta \, \in \, H^0(X,\, \text {ad}(E_G)\otimes K_X\otimes {{\mathcal {O}}}_X(D))$ such that $\theta (x)$ is compatible with the framing $\phi$ at x for every $x\, \in \, D$. We construct a holomorphic symplectic structure on the moduli space ${\mathcal {M}}_{FH}(G)$ of stable framed G–Higgs bundles on X. Moreover, we prove that the natural morphism from ${\mathcal {M}}_{FH}(G)$ to the moduli space ${\mathcal {M}}_{H}(G)$ of D-twisted G–Higgs bundles $(E_G,\, \theta )$ 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 $(G,\, \{H_x\}_{x\in D})$ is taken to be $(\text {GL}(r,{\mathbb C}),\, \{\text {I}_{r\times r}\}_{x\in D})$. We also investigate the Hitchin system for the moduli space ${\mathcal {M}}_{FH}(G)$ and its relationship with that for ${\mathcal {M}}_{H}(G)$.