Closed-string mirror symmetry for dimer models

For all punctured Riemann surfaces arising as mirror curves of toric Calabi--Yau threefolds, we show that their symplectic cohomology is isomorphic to the compactly supported Hochschild cohomology of the noncommutative Landau--Ginzburg model defined on the NCCR of the associated toric Gorenstein singularities. This mirror correspondence is established by analyzing the closed-open map with boundaries on certain combinatorially defined immersed Lagrangians in the Riemann surface, yielding a ring isomorphism. We give a detailed examination of the properties of this isomorphism, emphasizing its relationship to the singularity structure.