On Hochschild invariants of Landau–Ginzburg orbifolds
On Hochschild invariants of Landau–Ginzburg orbifolds
复制标题
论 Landau-Ginzburg 环折的 Hochschild 不变量
DOI：
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发表时间：
2017
影响因子：
1.5
通讯作者：
D. Shklyarov
中科院分区：
文献类型：
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作者：
D. Shklyarov
给出了一种计算与多项式及其有限交换对称群相关的曲线代数的Hochschild上同调和同调的杯积和帽积的方法。对于具有孤立临界点的多项式，该方法给出了乘积的完整描述。我们还对相应的等变矩阵分解范畴的结果进行了重新表述。在与Alexey Basalaev共同撰写的附录中，我们应用这些公式计算了一类简单但非平凡的所谓可逆LG orbillold模型的Hochschild上同调。由此得到的代数与已有的文献中关于LG镜像对称性的同构，命名为扭曲的Milnor/Jacobian代数。我们推测，这对所有可逆LG模型都是成立的。在附录的第二部分中，这些公式被应用于另一类LG orborold，它已经出现在同调镜像对称的背景下，作为亏格2和更高的曲面的镜伴的一般类型的变种。结合曲面的同调镜像对称定理，我们的计算给出了曲面的Fukaya范畴的Hochschild上同调作为代数同构于曲面上同调的一个新的证明。
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.