Logarithmic CFT at generic central charge: from Liouville theory to the Qstate Potts model
Abstract
Using derivatives of primary fields (null or not) with respect to the conformal dimension, we build infinite families of nontrivial 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, normalizationindependent redefinition of the logarithmic coupling. In the example of the vacuum module at central charge zero, this parameter characterizes a Jordan block of dimension $3$, and takes the value $\frac{1}{48}$. We compute the corresponding nonchiral conformal blocks, although they in general do not satisfy any nontrivial differential equation. We show that these blocks appear in limits of Liouville theory fourpoint functions. As an application, we describe the logarithmic structures of the critical twodimensional $O(n)$ and $Q$state Potts models at generic central charge. The validity of our description is demonstrated by semianalytically bootstrapping fourpoint connectivities in the $Q$state Potts model to arbitrary precision. Moreover, we provide numerical evidence for the DelfinoViti conjecture for the threepoint connectivity. Our results hold for generic values of $Q$ in the complex plane and beyond.
 Publication:

SciPost Physics
 Pub Date:
 January 2021
 DOI:
 10.21468/SciPostPhys.10.1.021
 arXiv:
 arXiv:2007.04190
 Bibcode:
 2021ScPP...10...21N
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 38 pages, v5: corrected the value of the c=0 vacuum module's logarithmic coupling