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 Jac$(f,G)$ of $(f,G)$ defined by [2] is isomorphic as a $\mathbb{Z}/2\mathbb{Z}Z$-graded algebra to the Hochschild cohomology ${\mathsf{HH}}^{\ast }({\mathrm{MF}}_G(f))$ of the dg-category ${\mathrm{MF}}_G(f)$ of $G$-equivariant matrix factorizations of $f$ by calculating the product formula of ${\mathsf{HH}}^{\ast }({\mathrm{MF}}_G(f))$ given by Shklyarov [10]. We also discuss the relation of our previous results to the categorical equivalence.