AI 中文总结
研究带端点环面平方的部分包裹富卡娅范畴的霍赫希尔德上同调，通过显式生成集和带关系箭图描述dg自同态代数，根据端点数量判断形式性，利用相关方法计算上同调并构造dg变形。
AI 中文摘要
我们计算了带端点的环面平方的部分包裹富卡娅范畴的霍赫希尔德上同调。利用该范畴中的一个显式生成集，通过一个带关系的箭图给出了其dg自同态代数$\widetilde{\mathcal{A}}_{n_1,n_2}$的描述。我们表明，当一个边界分量有单个端点时，dg代数是形式的；然而，当两个边界都至少包含两个端点时，它不是形式的，其最小$A_\infty$模型带有非平凡运算$m_3$。这使我们能够通过约化系统和谱序列计算其霍赫希尔德上同调，并构造与所得霍赫希尔德上闭链相关的一族dg变形。
英文摘要
We compute the Hochschild cohomology of the partially wrapped Fukaya category of the symmetric square of an annulus with stops. Using an explicit generating set in this category, we give a description of its dg endomorphism algebra $\widetilde{\mathcal{A}}_{n_1,n_2}$ via a quiver with relations. We show that when one boundary component has a single stop, the dg algebra is formal; however, when both boundaries contain at least two stops, it is not formal, and its minimal $A_\infty$-model carries a nontrivial operation $m_3$. This allows us to compute its Hochschild cohomology via reduction systems and spectral sequences, and to construct a family of dg deformations associated to the resulting Hochschild cocycles.
Comments43 pages, 20 figures, comments are very welcome!
