Using derivatives of primary fields (null or not) with respect to the conformal dimension, we build infinite families of non-trivial logarithmic representations of the conformal algebra at generic central charge, with Jordan blocks of dimension $2$ or $3$. Each representation comes with one free parameter, which takes fixed values under assumptions on the existence of degenerate fields. This parameter can be viewed as a simpler, normalization-independent redefinition of the logarithmic coupling. We compute the corresponding non-chiral conformal blocks, and show that they appear in limits of Liouville theory four-point functions. As an application, we describe the logarithmic structures of the critical two-dimensional $O(n)$ and $Q$-state Potts models at generic central charge. The validity of our description is demonstrated by semi-analytically bootstrapping four-point connectivities in the $Q$-state Potts model to arbitrary precision. Moreover, we provide numerical evidence for the Delfino--Viti conjecture for the three-point connectivity. Our results hold for generic values of $Q$ in the complex plane and beyond.

中文翻译：

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通用中央收费的对数CFT：从Liouville理论到$ Q $状态的Potts模型

使用共形维数的主场的导数（无论是否为零），我们在通用中心电荷处构造无穷大的共形代数的非平凡对数表示形式，其中约旦块的尺寸为$ 2 $或$ 3 $。每个表示带有一个自由参数，在假设简并字段存在的情况下，该参数取固定值。可以将此参数视为对数耦合的更简单，独立于归一化的重新定义。我们计算了相应的非手性共形嵌段，并表明它们出现在Liouville理论四点函数的极限中。作为一个应用程序，我们以通用中心收费描述了关键的二维$ O（n）$和$ Q $状态的Potts模型的对数结构。我们的描述的正确性通过在$ Q $状态的Potts模型中半解析地引导四点连通性到任意精度来证明。此外，我们为三点连通性的Delfino-Viti猜想提供了数值证据。我们的结果适用于复杂平面及更高平面中的$ Q $通用值。