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 \({\mathcal {M}}_{H}(G)\).

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Acknowledgements

Remarks 2.12.3 and 2.5 are due to the referee. We are very grateful to the referee for these and other helpful comments. The first-named author thanks Centre de Recherches Mathématiques, Montreal, for hospitality. He is partially supported by a J. C. Bose Fellowship.

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Biswas, I., Logares, M. & Peón-Nieto, A. Moduli Spaces of Framed G–Higgs Bundles and Symplectic Geometry. Commun. Math. Phys. 376, 1875–1908 (2020). https://doi.org/10.1007/s00220-019-03531-3

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