We construct a new family of flat connections generalising the KZ connection, the Casimir connection and the dynamical connection. These new connections are attached to simply-laced graphs, and are obtained via quantisation of time-dependent Hamiltonian systems controlling the isomonodromic deformations of meromorphic connections on the sphere.

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.

We will define and study some generalisations of pure g-braid groups that occur in the theory of connections on curves, for any complex reductive Lie algebra g. They make up local pieces of the wild mapping class groups, which are fundamental groups of (universal) deformations of wild Riemann surfaces, underlying the braiding of Stokes data and generalising the usual mapping class groups. We will establish a general product decomposition for the local wild mapping class groups, and in many cases define a fission tree controlling this decomposition. Further in type A we will show one obtains cabled versions of braid groups, related to braid operads.

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