A topological algorithm for the Fourier transform of Stokes data at infinity

We give a topological description of the behaviour of Stokes matrices under the Fourier transform from infinity to infinity in a large number of cases of one level. This explicit, algorithmic statement is obtained by building on a recent result of T. Mochizuki about the Fourier transform of Stokes data of irregular connections on the Riemann sphere and by using the language of Stokes local systems due to P. Boalch. In particular, this induces explicit isomorphisms between wild character varieties, in a much larger range of examples than those for which such isomorphisms have previously been written down. We conjecture that these isomorphisms are compatible with the quasi-Hamiltonian structure on the wild character varieties.