论文标题
四点障碍的模块化练习 - 我
Modular Exercises for Four-Point Blocks -- I
论文作者
论文摘要
（理性）顶点操作员代数（VOAS）和2D共形场理论（CFTS）的圆环字符和圆环分区功能的众所周知的模块化特性一直是研究这类理论的宝贵工具。在这项工作中，我们证明，有理voas的四点手性块是$γ（2）$，$γ_0（2）$或$ \ mathrm {sl} _2 _2（\ mathbb {z}）$的矢量值模块形式。此外，我们证明，将四点相关器结合在一起，结合了圆锥形和抗塑形性手性手性块是模块化的不变性。特别是，在这种语言中，交叉对称性只是模块化的对称性。这提供了利用可用技术和有关模块化形式的知识，以确定或限制物理上有趣的数量，例如手性块和融合系数，我们用一些示例说明了这一点。我们还强调了一个球体通信的存在，将某些理论的球数等同于$ {\ Mathcal t} _s $的球体数量与另一个理论家族的圆环数量$ {\ Mathcal t} _t $。伴侣论文将深入研究更多的例子，并更系统地探索这个球体双重性。
The well-known modular property of the torus characters and torus partition functions of (rational) vertex operator algebras (VOAs) and 2d conformal field theories (CFTs) has been an invaluable tool for studying this class of theories. In this work we prove that sphere four-point chiral blocks of rational VOAs are vector-valued modular forms for the groups $Γ(2)$, $Γ_0(2)$, or $\mathrm{SL}_2(\mathbb{Z})$. Moreover, we prove that the four-point correlators, combining the holomorphic and anti-holomorphic chiral blocks, are modular invariant. In particular, in this language the crossing symmetries are simply modular symmetries. This gives the possibility of exploiting the available techniques and knowledge about modular forms to determine or constrain the physically interesting quantities such as chiral blocks and fusion coefficients, which we illustrate with a few examples. We also highlight the existence of a sphere-torus correspondence equating the sphere quantities of certain theories ${\mathcal T}_s$ with the torus quantities of another family of theories ${\mathcal T}_t$. A companion paper will delve into more examples and explore more systematically this sphere-torus duality.
