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

TL;DR

This paper provides a comprehensive topological algorithm to compute the Fourier transform of Stokes data for irregular connections on the Riemann sphere using Mochizuki's framework and Boalch's Stokes local systems. By interpreting the Legendre transform as a homeomorphism on exponential circles and employing deformation data on a fission surface, it yields explicit algebraic isomorphisms between wild character varieties on either side of the Fourier transform, with conjectured compatibility with quasi-Hamiltonian structures. The authors demonstrate the method in concrete Gaussian-type, Airy, and ramified z^{5/3}-w^{5/2} examples, showing that the transform can be understood entirely topologically and that the induced maps preserve or relate the symplectic geometry of the moduli spaces. Overall, the work broadens the class of cases with explicit Fourier-transform descriptions of Stokes data and lays groundwork for deeper links to symplectic and cluster-theoretic structures.

Abstract

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.

Figures (15)

• Figure 1: A local system $V^0\to\partial$ graded by a single Stokes circle $\langle q \rangle$, here with ramification order $r=3$. For any direction $d\in \partial$, the fibre $\langle q \rangle_d=\pi^{-1}(d)=\{k_0, \dots, k_{r-1}\}$ contains $r$ points, and we have a locally constant decomposition $V^0_d=\bigoplus_{i=0}^{r-1}V^0_d(k_i)$. It will often be convenient to view $V^0$ as a local system on the circle $\langle q \rangle$, in such a way that its fibre over the point $k_i\in \langle q \rangle$ is the graded piece $V^0_d(k_i)$.
• Figure 2: Stokes diagrams for two examples of irregular classes. The diagram represents the growth rate of the exponents as a function of the direction around the singularity. The dotted circle separates the regions where the exponential is growing or decreasing near $\infty$. The singular directions are the directions where we find a maximal distance between two strands, and the corresponding Stokes arrows are drawn. The Stokes directions are the directions where some of the strands of the diagram cross.
• Figure 3: Local picture: The halo and the tangential punctures at a singularity.
• Figure 4: The paths for expressing the monodromies of a Stokes local system.
• Figure 5: A deformation datum associated to a Stokes arrow over a singular direction can be pictured as a path from a leaf to another. Note that at the boundary the surface is equipped with a covering by the exponents, so it is best to picture the blue and red circles above the black one (in the third dimension).

Theorems & Definitions (51)

• Theorem 1.1: see Theorem \ref{['thm:trafoRule']} for the fully detailed statement
• Theorem 1.2: see Theorem \ref{['thm:algebraic']}
• Definition 2.1
• Definition 2.2
• Theorem 2.3: see Mal91
• Theorem 2.4
• Lemma 2.5
• proof
• Definition 3.1
• Definition 3.2