Topological computation of some Stokes phenomena on the affine line

TL;DR

This paper provides a topological method to compute Stokes phenomena for certain algebraic D-modules on the affine line, using Borel-Moore cycles, and extends to irregular cases like the Airy equation.

Contribution

It offers a purely topological approach to determine Stokes multipliers, recovering Malgrange's algebraic results through Borel-Moore cycles and irregular D-modules analysis.

Findings

Topological computation of Stokes multipliers using Borel-Moore cycles.

Recovery of Malgrange's algebraic results via topological methods.

Application to irregular D-modules like the Airy equation.

Abstract

Let $\mathcal M$ be a holonomic algebraic $\mathcal D$-module on the affine line, regular everywhere including at infinity. Malgrange gave a complete description of the Fourier-Laplace transform $\widehat{\mathcal M}$, including its Stokes multipliers at infinity, in terms of the quiver of $\mathcal M$. Let $F$ be the perverse sheaf of holomorphic solutions to $\mathcal M$. By the irregular Riemann-Hilbert correspondence, $\widehat{\mathcal M}$ is determined by the enhanced Fourier-Sato transform $F^\curlywedge$ of $F$. Our aim here is to recover Malgrange's result in a purely topological way, by computing $F^\curlywedge$ using Borel-Moore cycles. In this paper, we also consider some irregular $\mathcal M$'s, like in the case of the Airy equation, where our cycles are related to steepest descent paths.