| Résume | I'll describe some basic examples of braid group actions on moduli spaces that motivated the definition of wild Riemann surface and wild mapping class group. It explains how all the (cabled) G-braid groups appear naturally in 2d gauge theory, and the simplest case underlies the G-braid group action on the Drinfeld-Jimbo quantum group.
Plan:
- Hurwitz action as B_n action on tame genus zero character varieties
- quandle lifting of 1) for complex reflections
- Fourier dual of 1) via 2): braiding of Stokes matrices. Example: braiding of BPS states chez Cecotti-Vafa.
- G-version of 3) yielding all the G-braid groups.
- big picture and recent work: Definition of wild Riemann surfaces, wild character varieties, and wild mapping class groups.
Some References for 2)-5):
- 2),3) From Klein to Painlevé via Fourier, Laplace and Jimbo
Proc. London Math. Soc. (3) 90 (2005) 167–208
- 4) G-bundles, Isomonodromy and Quantum Weyl Groups
Int. Math. Res. Not. 22 (2002) 1129–1166
- 5) Geometry and braiding of Stokes data; Fission and wild character varieties
Annals of Math. 179 (2014) 301–365
- Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids (avec J. Douçot et G. Rembado)
Publ. R.I.M.S. 61 (2025) no. 3, 391–445
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