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

### Submission summary

 As Contributors: Rongvoram Nivesvivat · Sylvain Ribault Arxiv Link: https://arxiv.org/abs/2007.04190v3 (pdf) Code repository: https://gitlab.com/s.g.ribault/Bootstrap_Virasoro/-/tree/precision Date accepted: 2021-01-21 Date submitted: 2020-12-03 11:45 Submitted by: Nivesvivat, Rongvoram Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Mathematical Physics Approach: Theoretical

### 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. 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.

Published as SciPost Phys. 10, 021 (2021)

### List of changes

added clarifications and references following reviewers' suggestions, in particular in Section 2.2

### Submission & Refereeing History

#### Published as SciPost Phys. 10, 021 (2021)

Resubmission 2007.04190v3 on 3 December 2020
Submission 2007.04190v2 on 22 September 2020

## Reports on this Submission

### Anonymous Report 8 on 2021-1-12 (Contributed Report)

• Cite as: Anonymous, Report on arXiv:2007.04190v3, delivered 2021-01-12, doi: 10.21468/SciPost.Report.2397

### Report

The work is good for newcomers to the field. I think this version can be published.

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### Anonymous Report 7 on 2021-1-11 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2007.04190v3, delivered 2021-01-11, doi: 10.21468/SciPost.Report.2391

### Report

In the revised version the authors took into account in a detailed way referee comments concerning the logarithmic CFT methods, mostly about references to previous works. What makes the paper of considerable interest is the ability of the authors to develop those methods and to make them applicable to the challenging case of the conformal bootstrap for the Q-state Potts model and percolation. The difficulty of the problem also requires the combination of analytical and numerical methods. The authors manage to progress in both directions and to obtain new results. I confirm my recommendation for publication.

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### Report

The points raised in my previous reports have been addressed, so I recommend this manuscript for publication in SciPost Physics.

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