Quantum cohomology and Fukaya summands from monotone Lagrangian tori

TL;DR

This paper establishes a link between the quantum cohomology of a symplectic manifold and the superpotential of a monotone Lagrangian torus, confirming mirror symmetry predictions and analyzing Fukaya category summands.

Contribution

It introduces a decomposition of quantum cohomology via the closed-open map related to the superpotential, and verifies mirror symmetry expectations for monotone Lagrangian tori.

Findings

Decomposition of quantum cohomology into factors related to superpotential critical points.

Verification of mirror symmetry predictions for Fukaya category summands.

Constraints on superpotentials of monotone Lagrangian tori.

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 $\operatorname{QH}^*(X)$ into a product, where one factor is the localisation of the Jacobian ring $\operatorname{Jac} W_L$ 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