| Summary: | 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.
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