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.

McKay correspondence and orbifold equivalence

“…we take G = Z/2Z acting by (u, v) → (−u, −v) so that f is of type A 2b−1 and the singularity of f is of type D b+1 . This was observed in [2], where it is also shown that one can chose a suitable affine chart on Y isomorphic to the affine space. The matrix factorization constructed in loc.…”

“…In this case f is of type A 2 × A 2 and the singularity of f is of type D 4 . This was observed in [2].…”

“…The observation of this sort were made in [1], which was a motivation for the present paper. It was noticed there that several of the known examples of orbifold equivalence fit into this setting.…”

Mirror map for Fermat polynomial with non-abelian group of symmetries

“…In particular, the G-action does not mix up the sectors of A ′ f,G . For f being invertible polynomial the corresponding FJRW and Hochschild cohomology rings were computed in [FJJS,BT2,BTW16,BTW17].…”

Hochschild cohomology of Fermat type polynomials with non-abelian symmetries

“…For G ⊆ G d f the Frobenius algebra of (f, G) was considered in [K03,K09,BTW16,BTW17]. All these publications made an attempt to construct a new object employing some essential ideas -it was widely agreed how the vector space of the pair (f, G)…”