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

2023/08/25 by Daniel Pomerleano, Paul Seidel, Pomerleano, Daniel +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic structures and combinatorial models #Connection (principal bundle) #Divisor (algebraic geometry) #Geometry #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Noncommutative geometry #Pure mathematics #Symplectic geometry

paper · pdf · doi:10.48550/arxiv.2308.13567

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2023/08/25 · openalex created_date 2023/08/31 · openalex updated_date 2026/07/28

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.

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