The pre-symplectic geometry of opers and the holonomy map


Given a connected complex semi-simple Lie group G and a Riemann surface X, a G-oper on X is a higher rank generalization of a complex projective structure on X. These objects play an important role in integrable systems and geometric representation theory, a status that was cemented by the seminal work of Beilinson-Drinfeld \citeBD91.
For G a connected complex simple Lie group of adjoint type, we study the global deformation theory of G-opers on a connected, closed, oriented smooth surface Σof genus at least two. We exhibit the deformation space of G-opers on Σas a holomorphic fiber bundle over Teichmüller space, and elucidate the relationship with the deformation space of complex projective structures. Then, we show that there is a family of identifications of the deformation space of G-opers with a holomorphic vector bundle \mathcalB_G(Σ) over Teichmüller space whose typical fiber over a Riemann surface X is a sum of spaces of pluri-canonical sections.
Finally, we show that the holonomy map from the deformation space of G-opers to the deformation space of flat G-bundles on Σis a holomorphic immersion. As a consequence of this result, we show that the deformation space of G-opers carries a (pre-symplectic) closed holomorphic differential 2-form of constant rank, and we prove that a sub-family of the identifications of \mathcalB_G(Σ) with the deformation space of G-opers is a holomorphic pre-symplectic map for a natural holomorphic pre-symplectic form on \mathcalB_G(Σ). These results generalize the fundamental features of the deformation space of complex projective structures on Σto the setting of G-opers.


Research Groups: Groups and Geometry

Publication type: Journal

Journal: arXiV Preprint

Citation: arXiv:1804.04716 [math.DG]

Date Published: 2018


Registered Mode: imported from a bibtex file

Author: Andrew Sanders

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Created: 30th Jan 2020 at 09:40

Last updated: 5th Mar 2024 at 21:24

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