Logarithmic operators in $c=0$ bulk CFTs

TL;DR

This paper investigates the structure of logarithmic operators in $c=0$ bulk conformal field theories, revealing complex Jordan block formations and non-vanishing four-point functions that challenge previous assumptions.

Contribution

It provides a generic construction of logarithmic operators at $c=0$, analyzes their conformal data, and uncovers the role of higher-rank Jordan blocks in bulk correlation functions.

Findings

Four-point energy correlator does not vanish at $c=0$

Logarithmic operators form high-rank Jordan blocks

Zero-norm operators influence long-range correlations

Abstract

We study Kac operators (e.g. energy operator) in percolation and self-avoiding walk bulk CFTs with central charge $c=0$. The proper normalizations of these operators can be deduced at generic $c$ by requiring the finiteness and reality of the three-point constants in cluster and loop model CFTs. At $c=0$, Kac operators become zero-norm states and the bottom fields of logarithmic multiplets, and comparison with $c<1$ Liouville CFT suggests the potential existence of arbitrarily high rank Jordan blocks. We give a generic construction of logarithmic operators based on Kac operators and focus on the rank-2 pair of the energy operator mixing with the hull operator. By taking the $c\to 0$ limit, we compute some of their conformal data and use this to investigate the operator algebra at $c=0$. Based on cluster decomposition, we find that, contrary to previous belief, the four-point correlation function of the bulk energy operator does not vanish at $c=0$, and a crucial role is played by its coupling to the rank-3 Jordan block associated with the second energy operator. This reveals the intriguing way zero-norm operators build long-range higher-point correlations through the intricate logarithmic structures in $c=0$ bulk CFTs.