类 $\mathcal{S}$ 的扭曲手性代数与混合 Feigin-Frenkel 粘合
Twisted chiral algebras of class $\mathcal{S}$ and mixed Feigin-Frenkel gluing
AI总结：
本文针对非 simply laced 情形下类 $\mathcal{S}$ 手性代数与四维理论对应不清的问题，在 Arakawa 构造基础上引入外自同构扭转线，提出了类 $\mathcal{S}$ 手性代数的新定义并通过一致性检验。
摘要：
四维 $\mathcal{N}=2$ 超共形场论与顶点算子代数之间的对应关系，当应用于类 $\mathcal{S}$ 理论时，产生了一个丰富的顶点算子代数族，这些代数被称为类 $\mathcal{S}$ 的手性代数。Tomoyuki Arakawa 在 arXiv:1811.01577 中提出了这些顶点算子代数的一个非常统一的构造。arXiv:1811.01577 的构造以一个简单李代数 $\mathfrak{g}$ 的选择作为输入，并且无论 $\mathfrak{g}$ 是 simply laced 还是非 simply laced，都同样适用。然而，在非 simply laced 情形下，所得的顶点算子代数并不以任何明确的方式对应于已知的四维理论。另一方面，涉及非 simply laced 对称代数的类 $\mathcal S$ 理论的标准实现需要引入外自同构扭转线，这要求进一步发展 arXiv:1811.01577 的方法。在本文中，我们阐述了这些进一步发展，并提出了大多数带有外自同构扭转线的类 $\mathcal S$ 手性代数的定义。我们证明了我们的定义通过了一些一致性检验，并指出了一些重要的开放问题。
英文摘要：
The correspondence between four-dimensional $\mathcal{N}=2$ superconformal field theories and vertex operator algebras, when applied to theories of class $\mathcal{S}$, leads to a rich family of VOAs that have been given the monicker chiral algebras of class $\mathcal{S}$. A remarkably uniform construction of these vertex operator algebras has been put forward by Tomoyuki Arakawa in arXiv:1811.01577. The construction of arXiv:1811.01577 takes as input a choice of simple Lie algebra $\mathfrak{g}$, and applies equally well regardless of whether $\mathfrak{g}$ is simply laced or not. In the non-simply laced case, however, the resulting VOAs do not correspond in any clear way to known four-dimensional theories. On the other hand, the standard realisation of class $\mathcal S$ theories involving non-simply laced symmetry algebras requires the inclusion of outer automorphism twist lines, and this requires a further development of the approach of arXiv:1811.01577. In this paper, we give an account of those further developments and propose definitions of most chiral algebras of class $\mathcal S$ with outer automorphism twist lines. We show that our definition passes some consistency checks and point out some important open problems.
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