Quantum cohomology and Fukaya summands from monotone Lagrangian tori

Jack Smith

doi:10.5802/jep.290

Quantum cohomology and Fukaya summands from monotone Lagrangian tori
[Cohomologie quantique et facteurs de Fukaya à partir de tores lagrangiens monotones]

Jack Smith ¹

¹ St John’s College, Cambridge, CB2 1TP, United Kingdom

Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 287-318.

Résumé
Abstract

Soit $L$ un tore lagrangien monotone dans une variété symplectique compacte $X$ , de super-potentiel $W_{L}$ . Nous montrons qu’une application ouverte-fermée définie géométriquement induit une décomposition de la cohomologie quantique ${QH}^{*} (X)$ en un produit où l’un des facteurs est la localisation de l’anneau jacobien $Jac W_{L}$ en l’ensemble des points critiques isolés de $W_{L}$ . La démonstration fait intervenir la description des facteurs de la catégorie de Fukaya correspondant à ce facteur — vérifiant ainsi ce que prédit la symétrie miroir — et l’introduction d’un critère de génération automatique, à la manière de Ganatra et Sanda, qui peut avoir son intérêt propre. Nous appliquons nos résultats pour comprendre la structure de la cohomologie quantique et pour obtenir des contraintes sur les super-potentiels possibles des tores monotones.

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.

Reçu le : 2024-10-15
Accepté le : 2025-02-07
Publié le : 2025-02-18

DOI : 10.5802/jep.290

Classification : 53D45, 53D37, 53D12
Keywords: Quantum cohomology, Fukaya category, Lagrangian torus, homological smoothness
Mots-clés : Cohomologie quantique, catégorie de Fukaya, tore lagrangien, lissité homologique

Affiliations des auteurs :

Jack Smith ¹

¹ St John’s College, Cambridge, CB2 1TP, United Kingdom

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Jack Smith. Quantum cohomology and Fukaya summands from monotone Lagrangian tori. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 287-318. doi : 10.5802/jep.290. https://jep.centre-mersenne.org/articles/10.5802/jep.290/

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