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.

中文翻译：

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关于 Landau-Ginzburg orbifolds 的 Hochschild 不变量

我们开发了一种计算与多项式及其有限阿贝尔对称群相关的弯曲代数的 Hochschild 上同调和同调的杯和帽积的方法。对于具有孤立临界点的多项式，该方法会生成对产品的完整描述。我们还重新制定了相应类别的等变矩阵分解的结果。在与 Alexey Basalaev 共同编写的附录中，我们应用公式来计算一个简单但非平凡的所谓可逆 LG 轨道模型的 Hochschild 上同调。由此产生的代数与在 LG 镜像对称性文献中已经出现的以扭曲或折叠的米尔诺/雅可比代数为名的代数是同构的。我们推测这适用于所有可翻转的 LG 型号。在附录的第二部分，这些公式被应用于不同类别的 LG 轨道，这些轨道出现在同调镜像对称的背景下，用于作为 2 类及更高版本表面的镜像伙伴的一般类型的变种。结合表面的同调镜像对称定理，我们的计算得出了一个新的证明，即表面的 Fukaya 范畴的 Hochschild 上同调作为代数与表面的上同调同构。