Quantum cohomology and Fukaya summands from 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