The quantum connection, Fourier-Laplace transform, and families of A-infinity-categories

TL;DR

This paper studies the quantum connection on symplectic manifolds with divisors, revealing its singularity at infinity as unramified exponential type through categorical and geometric methods.

Contribution

It introduces a new categorical interpretation of the Fourier-Laplace transform and proves the regularity of the quantum connection's singularity at infinity.

Findings

Quantum connection has regular singularity at zero.

Singularity at infinity is of unramified exponential type.

Categorical approach links symplectic cohomology and Fukaya categories.

Abstract

Take a closed monotone symplectic manifold containing a smooth anticanonical divisor. The quantum connection on its cohomology has singularities at zero and infinity (in the quantum parameter). At zero it has a regular singular point, by definition. We show that the singularity at infinity is of unramified exponential type. The argument involves: realizing cohomology as a deformation of the symplectic cohomology of the divisor complement; the corresponding deformation of the wrapped Fukaya category; a new categorical interpretation of the Fourier-Laplace transform of D-modules; and the regularity theorem of Petrov-Vaintrob-Vologodsky in noncommutative geometry.