论文标题
超符号指数的模块化分解
Modular factorization of superconformal indices
论文作者
论文摘要
四维$ \ mathcal {n} = 1 $量规理论的超符合索引将其分解为holomorphic块。我们将其解释为一个模块化属性，该属性是由$ sl（3，\ mathbb {z}）$和$ sl（2，\ mathbb {z}）\ ltimes \ mathbb {z}^2 $转换产生的。前者对应于胶合的转化，后者对应于整体大的差异性，这既与基础几何形状的Heegaard分裂有关。对更一般的转换的扩展使我们争辩说，可以根据模块化（一致性子）组参数构元的全体形态块家族来分解给定指数。我们发现该提案与椭圆形$γ$函数的新模块化属性之间的确切一致性。这导致了我们对一般$ \ MATHCAL {N} = 1 $量学理论的超符合镜头索引的``模块化分解''的猜想。我们在自由手性多重和SQED的背景下为猜想提供了证据，并勾勒出我们的论点扩展到更通用的规定理论。基于此结果，我们系统地证明了超符合镜头指数的归一化部分定义了与$ sl（3，\ Mathbb {z}）$相关的非平凡的第一同居类。最后，我们使用此框架为通用镜头空间指数提出了一个公式。
Superconformal indices of four-dimensional $\mathcal{N}=1$ gauge theories factorize into holomorphic blocks. We interpret this as a modular property resulting from the combined action of an $SL(3,\mathbb{Z})$ and $SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2$ transformation. The former corresponds to a gluing transformation and the latter to an overall large diffeomorphism, both associated with a Heegaard splitting of the underlying geometry. The extension to more general transformations leads us to argue that a given index can be factorized in terms of a family of holomorphic blocks parametrized by modular (congruence sub)groups. We find precise agreement between this proposal and new modular properties of the elliptic $Γ$ function. This leads to our conjecture for the ``modular factorization'' of superconformal lens indices of general $\mathcal{N}=1$ gauge theories. We provide evidence for the conjecture in the context of the free chiral multiplet and SQED, and sketch the extension of our arguments to more general gauge theories. Based on this result, we systematically prove that a normalized part of superconformal lens indices defines a non-trivial first cohomology class associated with $SL(3,\mathbb{Z})$. Finally, we use this framework to propose a formula for the general lens space index.
