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 $\mathbb Z/2\mathbb ZZ$-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 \[10]. We also discuss the relation of our previous results to the categorical equivalence.