On Hochschild invariants of Landau-Ginzburg orbifolds

Abstract

We develop an approach to calculating the cup and cap products on Hochschild cohomology and homology of curved algebras associated with polynomials and their finite abelian symmetry groups. For polynomials with isolated critical points, the approach yields a complete description of the products. We also reformulate the result for the corresponding categories of equivariant matrix factorizations. In an Appendix written jointly with Alexey Basalaev, we apply the formulas to calculate the Hochschild cohomology of a simple but non-trivial class of so-called invertible LG orbifold models. The resulting algebras turn out to be isomorphic to what has already appeared in the literature on LG mirror symmetry under the name of twisted or orbifolded Milnor/Jacobian algebras. We conjecture that this holds true for all invertible LG models. In the second part of the Appendix, the formulas are applied to a different class of LG orbifolds which have appeared in the context of homological mirror symmetry for varieties of general type as mirror partners of surfaces of genus 2 and higher. In combination with a homological mirror symmetry theorem for the surfaces, our calculation yields a new proof of the fact that the Hochschild cohomology of the Fukaya category of a surface is isomorphic, as an algebra, to the cohomology of the surface.