From hypertoric geometry to bordered Floer homology via the $$m=1$$ amplituhedron

Selecta Mathematica Pub Date : 2024-04-03 DOI:10.1007/s00029-024-00932-8

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Abstract

We relate the Fukaya category of the symmetric power of a genus zero surface to deformed category $\mathcal {O}$ of a cyclic hypertoric variety by establishing an isomorphism between algebras defined by Ozsváth–Szabó in Heegaard–Floer theory and Braden–Licata–Proudfoot–Webster in hypertoric geometry. The proof extends work of Karp–Williams on sign variation and the combinatorics of the $m=1$ amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of $\mathfrak {gl}(1|1)$ , and generalize our isomorphism to give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement.

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通过 $$m=1$$ 放大系数从超几何到有边弗洛尔同调

摘要我们通过建立由 Ozsváth-Szabó 在 Heegaard-Floer 理论中定义的代数与 Braden-Licata-Proudfoot-Webster 在超几何中定义的代数之间的同构关系，将零属曲面的对称幂的 Fukaya 范畴与循环超几何的变形范畴 $\mathcal {O}$ 联系起来。证明扩展了卡普-威廉姆斯（Karp-Williams）关于符号变化和 $m=1$ 振子面体组合学的工作。然后，我们使用与循环排列相关的代数来构造 $\mathfrak {gl}(1|1)$ 的分类行动，并将我们的同构性加以推广。，并将我们的同构概括为对复化超平面补集的 Fukaya 范畴的猜想代数描述。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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