# Dimer Models and Hochschild Cohomology

## TL;DR

This paper computes the Hochschild cohomology of Jacobi algebras from dimer models on tori, revealing their Batalin--Vilkovisky structures and implications for noncommutative geometry and mirror symmetry.

## Contribution

It provides explicit Hochschild cohomology calculations for Jacobi algebras from zigzag consistent dimers, including their BV structures, linking combinatorics with homological mirror symmetry.

## Key findings

- Explicit Hochschild cohomology formulas for Jacobi algebras.
- Characterization of BV structures induced by Calabi--Yau properties.
- Computation of compactly supported Hochschild cohomology for matrix factorizations.

## Abstract

Dimer models provide a method of constructing noncommutative crepant resolutions of affine toric Gorenstein threefolds. In homological mirror symmetry, they can also be used to describe noncommutative Landau--Ginzburg models dual to punctured Riemann surfaces. For a zigzag consistent dimer embedded in a torus, we explicitly compute the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics. This includes a full characterization of the Batalin--Vilkovisky structure induced by the Calabi--Yau structure of the Jacobi algebra. We then compute the compactly supported Hochschild cohomology of the category of matrix factorizations for the Jacobi algebra with its canonical potential.