Logarithmic CFT at generic central charge: from Liouville theory to the $Q$-state Potts model

TL;DR

This paper constructs and analyzes infinite families of logarithmic conformal field theory representations at generic central charge, connecting Liouville theory and Potts models, and provides numerical validation of theoretical predictions.

Contribution

It introduces a new method to build logarithmic representations using derivatives of primary fields, and applies this to analyze critical Potts and O(n) models at arbitrary central charge.

Findings

Constructed logarithmic representations with Jordan blocks of dimension 2 or 3.

Computed non-chiral conformal blocks that appear in Liouville theory limits.

Validated the approach by bootstrapping four-point connectivities and supporting the Delfino--Viti conjecture.

Abstract

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. 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 non-chiral conformal blocks, although they in general do not satisfy any nontrivial differential equation. We show that these blocks 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.