带扭质量的红外代数研究
On the Algebra of the Infrared with Twisted Masses
AI总结：
本文将基于web的红外代数形式体系推广至带非平凡扭质量的$\nmathcal{N}=(2,2)$量子场论，针对单真空单扭质量理论推导相关构造，发现闭孤子Fock空间等新现象，并以实例演示该形式体系。
摘要：
红外代数（Algebra of the Infrared）\n参考文献：Gaiotto:2015aoa\n是一种仅利用BPS扇区的信息，构建二维有质量$\nmathcal{N}=(2,2)$理论的局域可观测量、界面以及超对称边界条件范畴的框架，所得框架被称为“基于web的形式体系”。本文启动了对基于web的形式体系的推广工作，使其涵盖比参考文献Gaiotto:2015aoa中讨论的范围更广的一类$\nmathcal{N}=(2,2)$量子场论：即带非平凡扭质量的理论。其中关键的新要素是固定真空扇区内存在BPS粒子。本文针对允许这类BPS粒子存在的最简单一类理论——即单真空、单扭质量的理论——推导了基于web的形式体系。我们表明，即便在这种简单设定下，也存在有趣的新现象，包括闭孤子Fock空间的出现以及Koszul对偶代数的自然显现。从数学角度看，研究带扭质量的理论包含研究由闭全纯1-形式定义的Landau-Ginzburg模型的A类边界条件的Fukaya-Seidel范畴。本文勾勒了针对具有单个Morse零点和单个非平凡周期的1-形式的A类边界条件范畴的基于web的构造。我们通过一个极具启发性的例子明确演示了我们的形式体系。
英文摘要：
The Algebra of the Infrared \cite{Gaiotto:2015aoa} is a framework to construct local observables, interfaces, and categories of supersymmetric boundary conditions of massive $\mathcal{N}=(2,2)$ theories in two dimensions by using information only about the BPS sector. The resulting framework is known as the ``web-based formalism.'' In this paper we initiate the generalization of the web-based formalism to include a much wider class of $\mathcal{N}=(2,2)$ quantum field theories than was discussed in \cite{Gaiotto:2015aoa}: theories with non-trivial twisted masses. The essential new ingredient is the presence of BPS particles within a fixed vacuum sector. In this paper we work out the web-based formalism for the simplest class of theories that allow for such BPS particles: theories with a single vacuum and a single twisted mass. We show that even in this simple setting there are interesting new phenomenon including the emergence of Fock spaces of closed solitons and a natural appearance of Koszul dual algebras. Mathematically, studying theories with twisted masses includes studying the Fukaya-Seidel category of A-type boundary conditions for Landau-Ginzburg models defined by a closed holomorphic one-form. This paper sketches a web-based construction for the category of A-type boundary conditions for one-forms with a single Morse zero and a single non-trivial period. We demonstrate our formalism explicitly in a particularly instructive example.
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