The orbifold Hochschild product for Fermat hypersurface

TL;DR

This paper studies the algebraic structure of the cohomology of polyvector fields on orbifolds formed by Fermat hypersurfaces quotiented by abelian groups, confirming associativity and mirror symmetry predictions.

Contribution

It provides new examples of associative product structures on orbifold cohomology for Fermat hypersurfaces and proves the conjectural isomorphism with Hochschild cohomology for Calabi-Yau cases.

Findings

Product structure is associative for Fermat hypersurface quotients.

Confirmed the conjectural isomorphism between polyvector field cohomology and Hochschild cohomology.

Bigrading matches predictions from homological mirror symmetry.

Abstract

Let $G$ be an abelian group acting on a smooth algebraic variety $X$. We investigate the product structure and the bigrading on the cohomology of polyvector fields on the orbifold $[X/G]$, as introduced by C\u{a}ld\u{a}raru and Huang. In this paper we provide many new examples given by quotients of Fermat hypersurfaces, where the product is shown to be associative. This is expected due to the conjectural isomorphism at the level of algebras between the cohomology of polyvector fields and Hochschild cohomology of orbifolds. We prove this conjecture for Calabi-Yau Fermat hypersurface orbifold. We also show that for Calabi-Yau orbifolds, the multiplicative bigrading on the cohomology of polyvector fields agrees with what is expected in homological mirror symmetry.