Topological Laplace Transform and Decomposition of nc-Hodge Structures

TL;DR

This paper introduces a topological Laplace transform for Stokes structures, demonstrating its compatibility with Fourier transforms of D-modules and exploring applications in non-commutative Hodge structures and Landau-Ginzburg models.

Contribution

It constructs a new topological Laplace transform functor that links Stokes structures to constructible sheaves and proves its compatibility with Fourier transforms of D-modules, with applications in non-commutative Hodge theory.

Findings

Establishes a functorial equivalence between Stokes structures and constructible sheaves.

Shows compatibility of the transform with Fourier transforms of D-modules.

Connects spectral decomposition of nc-Hodge structures to vanishing cycle decomposition.

Abstract

We construct the topological Laplace transform functor from Stokes structures of exponential type to constructible sheaves on $\mathbb C$ with vanishing cohomology. We show that it is compatible with the Fourier transform of $D$-modules, and induces an equivalence of categories. We give two applications of the construction. First, we study the Fourier transform of B-model nc-Hodge structures associated to Landau-Ginzburg models, and prove the compatibility between the $\mathbb Q$-structure and the Stokes structure from the connection. Second, we relate the spectral decomposition of nc-Hodge structures to the vanishing cycle decomposition after Fourier transform via choices of Gabrielov paths. This is motivated by the study of the atomic decomposition of A-model nc-Hodge structures associated to smooth projective varieties.