Twisted local G-wild mapping class groups

TL;DR

This work develops a comprehensive framework for twisted local wild mapping class groups arising from irregular/ramified isomonodromic deformations of connections on principal G-bundles over fixed pointed curves. It reveals that the deformation spaces decompose into products of complex hyperplane complements and are governed by Galois-centralizer data in Weyl groups, leading to natural actions by (twisted) complex reflection groups on wild character varieties. A key innovation is the introduction of fission trees, which classify the hyperplane arrangements for classical Lie types (A,B,C,D), including noncrystallographic phenomena in type D and interior twists, via a Springer-Lehrer-type lens. The paper also extends the theory to interior twists, relates the deformations to generalized root-valuation stratifications, and frames the results within the theory of reflection cosets (Howlett, Lehrer–Springer) and spetses, providing explicit descriptions in type A and BC and crystallographic classifications in type D. Overall, these results furnish a robust topological and combinatorial toolkit for understanding local pieces of wild Teichmüller-type spaces and their symplectic/Poisson actions, with potential applications to isomonodromic integrable systems and quantum representations.

Abstract

We consider the isomonodromic deformations of irregular-singular connections defined on principal bundles over complex curves: for any complex reductive structure group G, and any polar divisor; allowing for a twisted/ramified formal normal form at each pole, and for twists in the interior of the curve. (This covers the general case in 2-dimensional meromorphic gauge theory.) We focus on the irregular moduli of the connections, studying the fundamental groups of the spaces of admissible deformations of their irregular types/classes, i.e., the local wild mapping class groups in the title. To describe them, we first take the viewpoint of (nonsplit) reflections cosets in Springer/Lehrer--Springer theory, which yields in particular new modular interpretations of complex reflection groups -- and their braid groups. Then we introduce new `fission' trees to treat structure groups of any (simple) classical type, leading to a complete classification of the corresponding hyperplane arrangements, and singling out an infinite family of noncrystallographic examples in type D. Moreover, we reinterpret Bessis' lift of Springer theory as the study of `quasi-generic' deformations, corresponding to irregular singularities whose leading coefficient is regular semisimple upon pullback along a local cyclic covering of the base curve. Finally, we rephrase much of this material in terms of generalized root-valuation stratifications.

Figures (10)

• Figure 1: Combinations of vertex- and edge-type in a full $BC$-branch.
• Figure 2: Comparison between full branches in type $A$ and $BC$
• Figure 3: The (eight) different types of fission for two $BC$-Stokes circles.
• Figure 4: Type-$BC$ branchings. The branching vertex is dotted, indicating that its type is not prescribed by the branching type.
• Figure 5: Labelled fission tree of Exmp. \ref{['ex:example_fission_tree']}.

Theorems & Definitions (115)

• proof : Proof postponed to \ref{['proof:lem_primitive_irregular_types']}
• proof
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• proof
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• proof : Proof postponed to \ref{['proof:lem_about_setwise_stabilizers']}
• proof : Proof postponed to \ref{['proof:lem_about_pointwise_stabilizers']}
• proof
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• proof : Proof postponed to \ref{['proof:lem_no_marking']}