Dimer models and Hochschild cohomology

Dimer models have appeared in the context of noncommutative crepant resolutions and homological mirror symmetry for punctured Riemann surfaces. For a zigzag consistent dimer embedded in a torus, we explicitly describe the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics. We then compute the compactly supported Hochschild cohomology of the category of matrix factorizations for the Jacobi algebra with its canonical potential.