超环面空间的扭曲拟映射与辛对偶
Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces
AI总结：
本文研究三维阿贝尔超对称规范理论中超环面簇的扭曲拟映射模空间，证明其奇异上同调等同于量子化环路空间上完整模的Ext群，并利用辛对偶给出扭曲拟映射不变量生成函数的表达式及旗箭图情形的闭公式。
摘要：
我们研究到超环面簇 $X$ 的扭曲拟映射的模空间，该簇作为三维阿贝尔超对称规范理论的希格斯分支出现。这些模空间参数化一般的箭图表示，其构建块是 $\mathbb{P}^1$ 上秩一层的映射，并受到与箭图相关的稳定性条件的约束，该条件同时涉及层和映射。我们证明这些模空间的奇异上同调自然等同于 $X$ 的“量子化环路空间”上的一对完整模的 Ext 群，我们将该量子化环路空间视为具有无穷多个物质场的相关理论的希格斯分支。我们构造该理论的库仑分支，并发现它是与 $X$ 相关联的库仑分支的周期类比。利用辛对偶的形式化框架，我们推导出扭曲拟映射不变量的生成函数的表达式，该表达式以周期库仑分支上某个倾斜模的特征标表示。当 $X$ 作为 $N$ 步旗箭图的阿贝尔化出现时，我们给出了该生成函数的闭公式。
英文摘要：
We study moduli spaces of twisted quasimaps to a hypertoric variety $X$, arising as the Higgs branch of an abelian supersymmetric gauge theory in three dimensions. These parametrise general quiver representations whose building blocks are maps between rank one sheaves on $\mathbb{P}^1$, subject to a stability condition, associated to the quiver, involving both the sheaves and the maps. We show that the singular cohomology of these moduli spaces is naturally identified with the Ext group of a pair of holonomic modules over the `quantized loop space' of $X$, which we view as a Higgs branch for a related theory with infinitely many matter fields. We construct the coulomb branch of this theory, and find that it is a periodic analogue of the coulomb branch associated to $X$. Using the formalism of symplectic duality, we derive an expression for the generating function of twisted quasimap invariants in terms of the character of a certain tilting module on the periodic coulomb branch. We give a closed formula for this generating function when $X$ arises as the abelianisation of the $N$-step flag quiver.
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