类$\mathcal{S}_k$的椭圆量子曲线
Elliptic Quantum Curves of Class $\mathcal{S}_k$
摘要：
量子曲线源自与4d $\mathcal{N}=2$规范理论相关的Seiberg-Witten曲线，通过将坐标提升为非交换算子。这样，曲线的代数方程被解释为一个算子方程，其中哈密顿量作用于具有零本征值的波函数。我们发现，这种结构一般化时，考虑环面紧化6D $\mathcal{N}=(1,0)$ SCFT。相应的量子曲线在性质上是椭圆的，因此相关的本征向量/本征值可以用雅可比形式表示。在本文中，我们专注于类6d SCFT所产生的M5膜横向的$\mathbb{C}^2/\mathbb{Z}_k$奇异性。在紧化2-环面的尺寸为零的极限中，相应的四维$\mathcal{N}=2$理论被称为$\mathcal{S}_k$类。我们明确地表明，与量子曲线相关联的本征向量是余维2表面算符的期望值，而相应的本征值是余维4威尔逊表面期望值。
英文摘要：
Quantum curves arise from Seiberg-Witten curves associated to 4d $\mathcal{N}=2$ gauge theories by promoting coordinates to non-commutative operators. In this way the algebraic equation of the curve is interpreted as an operator equation where a Hamiltonian acts on a wave-function with zero eigenvalue. We find that this structure generalises when one considers torus-compactified 6d $\mathcal{N}=(1,0)$ SCFTs. The corresponding quantum curves are elliptic in nature and hence the associated eigenvectors/eigenvalues can be expressed in terms of Jacobi forms. In this paper we focus on the class of 6d SCFTs arising from M5 branes transverse to a $\mathbb{C}^2/\mathbb{Z}_k$ singularity. In the limit where the compactified 2-torus has zero size, the corresponding 4d $\mathcal{N}=2$ theories are known as class $\mathcal{S}_k$. We explicitly show that the eigenvectors associated to the quantum curve are expectation values of codimension 2 surface operators, while the corresponding eigenvalues are codimension 4 Wilson surface expectation values.
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