Quantum cohomology and Fukaya summands from monotone Lagrangian tori

2024/09/12 by Jack Smith, Smith, Jack
Mathematics · #53D12 #53D37 #53D45 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2409.07922

openalex publication_date 2024/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L be a monotone Lagrangian torus inside a compact symplectic manifold X, with superpotential W_L. We show that a geometrically-defined closed-open map induces a decomposition of the quantum cohomology QH^*(X) into a product, where one factor is the localisation of the Jacobian ring Jac W_L at the set of isolated critical points of W_L. The proof involves describing the summands of the Fukaya category corresponding to this factor -- verifying the expectations of mirror symmetry -- and establishing an automatic generation criterion in the style of Ganatra and Sanda, which may be of independent interest. We apply our results to understanding the structure of quantum cohomology and to constraining the possible superpotentials of monotone tori

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