论文标题
类$ \ mathcal {s} _k $的椭圆量量曲线
Elliptic Quantum Curves of Class $\mathcal{S}_k$
论文作者
论文摘要
量子曲线是由Seiberg-Witten曲线与4D $ \ MATHCAL {N} = 2 $量学理论相关的曲线，通过将坐标提升为非交通运算符。通过这种方式，曲线的代数方程被解释为算子方程，其中哈密顿量在带有零特征值的波功能上作用。我们发现，当人们认为圆环成熟的6D $ \ MATHCAL {n} =（1,0）$ scfts时，这种结构会泛化。相应的量子曲线本质上是椭圆形的，因此可以用雅各比形式表示相关的特征向量/特征值。在本文中，我们专注于M5 Branes横向到$ \ Mathbb {C}^2/\ Mathbb {z} _K $ singularity的6D SCFTS类。在压实2螺旋的限制中，相应的4D $ \ MATHCAL {n} = 2 $理论被称为$ \ Mathcal {S} _K $。我们明确表明，与量子曲线相关的特征向量是Codimension 2表面运算符的期望值，而相应的特征值为Codimension 4 Wilson表面期望值。
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.
