Dimer Models and Hochschild Cohomology
arXiv:1908.03005
The paper computes the Hochschild cohomology of Jacobi algebras arising from zigzag‑consistent dimer models on a torus, describes the induced Batalin‑Vilkovisky structure, and determines the compactly supported Hochschild cohomology of the associated matrix factorization category.
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.
46 pages