Topology of irregular isomonodromy times on a fixed pointed curve
Jean Douçot, Gabriele Rembado
TL;DR
This work develops an intrinsic, algebraic-topological framework for nongeneric irregular isomonodromy on a fixed pointed curve by introducing full/nonpure local wild mapping class groups $\Gamma_{\overline Q}$. It builds a structural theory around Weyl group fission, decomposing deformation spaces through Levi subalgebras and yielding subquotients $W_{\mathfrak g \mid \bm{\mathfrak h}}$ that govern the nonpure monodromy, with deformations organized along fission chains and ranked trees. In type A, the authors provide a complete description by identifying $\Gamma_{\overline Q}$ with a recursively defined cabled braid group $\mathscr B(T,\bm r)$ on ranked fission trees, and extend the approach to general simple types via automorphism/wreath constructions. The results connect the geometry of wild character varieties, Stokes data, and isomonodromy to explicit braid-type groups, laying groundwork for quantization, Poisson actions, and applications to irregular Painlevé-type systems. Overall, the paper furnishes a detailed, computable bridge from irregular isomonodromy times to generalized braid/cabled-braid group actions, enabling concrete analysis of monodromy representations and their symmetries.
Abstract
We will define and study (moduli) spaces of deformations of irregular classes on Riemann surfaces, which provide an intrinsic viewpoint on the `times' of irregular isomonodromy systems in general. Our aim is to study the deeper generalisation of the G-braid groups that occur as fundamental groups of such deformation spaces, with particular focus on the generalisation of the full G-braid groups.