Hochschild cohomology and orbifold Jacobian algebras associated to invertible polynomials

Abstract

Let $f$ be an invertible polynomial and $G$ a group of diagonal symmetries of $f$. This note shows that the orbifold Jacobian algebra $\mathrm{Jac}(f,G)$ of $(f,G)$ defined by the authors and Elisabeth Werner in arXiv:1608.08962 is isomorphic as a $\mathbb{ZZ}/2\mathbb{ZZ}$-graded algebra to the Hochschild cohomology $\mathsf{HH}^*(\mathrm{MF}_G(f))$ of the dg-category $\mathrm{MF}_G(f)$ of $G$-equivariant matrix factorizations of $f$ by calculating the product formula of $\mathsf{HH}^*(\mathrm{MF}_G(f))$ given by Shklyarov in arXiv:1708.06030. We also discuss the relation of our previous results to the categorical equivalence.