Given an arbitrary graph $\Gamma$ and non-negative integers $g_v$ for each vertex $v$ of $\Gamma$, let $X_\Gamma$ be the Weinstein $4$-manifold obtained by plumbing copies of $T^*\Sigma_v$ according to this graph, where $\Sigma_v$ is a surface of genus $g_v$. We compute the wrapped Fukaya category of $X_\Gamma$ (with bulk parameters) using Legendrian surgery extending our previous work arXiv:1502.07922 where it was assumed that $g_v=0$ for all $v$ and $\Gamma$ was a tree. The resulting algebra is recognized as the (derived) multiplicative preprojective algebra (and its higher genus version) defined by Crawley-Boevey and Shaw arXiv:math/0404186. Along the way, we find a smaller model for the internal DG-algebra of Ekholm-Ng arXiv:1307.8436 associated to $1$-handles in the Legendrian surgery presentation of Weinstein $4$-manifolds which might be of independent interest.

中文翻译：

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管道的深谷范畴和乘法前投影代数

给定任意图 $\Gamma$ 和 $\Gamma$ 的每个顶点 $v$ 的非负整数 $g_v$，让 $X_\Gamma$ 是通过对 $T^* 的管道副本获得的 Weinstein $4$-流形\Sigma_v$ 根据此图，其中 $\Sigma_v$ 是 $g_v$ 属的表面。我们使用 Legendrian 手术计算了 $X_\Gamma$ 的包裹 Fukaya 类别（具有批量参数）扩展了我们之前的工作 arXiv:1502.07922，其中假设所有 $v$ 和 $\Gamma$ 的 $g_v=0$ 是一棵树. 由此产生的代数被认为是由 Crawley-Boevey 和 Shaw arXiv:math/0404186 定义的（派生的）乘法前投影代数（及其更高的属版本）。在此过程中，我们为 Ekholm-Ng arXiv:1307 的内部 DG 代数找到了一个较小的模型。