Closed-string mirror symmetry for dimer models

TL;DR

This paper establishes a mirror symmetry correspondence between symplectic cohomology of punctured Riemann surfaces and Hochschild cohomology of noncommutative Landau-Ginzburg models, revealing deep links in toric Calabi-Yau geometry.

Contribution

It proves a new isomorphism connecting symplectic and Hochschild cohomologies for mirror curves of toric Calabi-Yau threefolds, using immersed Lagrangians and the closed-open map.

Findings

Ring isomorphism between symplectic and Hochschild cohomologies.

Analysis of the closed-open map with boundary conditions.

Relationship of the isomorphism to singularity structures.

Abstract

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.